History of Topics in Special Relativity/Lorentz transformation (general)

Most general Lorentz transformations

The general w:quadratic form q(x) with coefficients of a w:symmetric matrix A, the associated w:bilinear form b(x,y), and the w:linear transformations of q(x) and b(x,y) into q(x′) and b(x′,y′) using the w:transformation matrix g, can be written as[1]

{\displaystyle {\begin{matrix}{\begin{aligned}q=\mathbf {x} ^{\mathrm {T} }\cdot \mathbf {A} \cdot \mathbf {x} \end{aligned}}=q'=\mathbf {x} ^{\mathrm {\prime T} }\cdot \mathbf {A} '\cdot \mathbf {x} '\\b=\mathbf {x} ^{\mathrm {T} }\cdot \mathbf {A} \cdot \mathbf {y} =b'=\mathbf {x} ^{\mathrm {\prime T} }\cdot \mathbf {A} '\cdot \mathbf {y} '\\\left(\mathbf {A} =\mathbf {A} ^{\rm {T}}\right)\\\hline \left.{\begin{aligned}\mathbf {x} '&=\mathbf {g} \cdot \mathbf {x} \\\mathbf {x} &=\mathbf {g} ^{-1}\cdot \mathbf {x} '\end{aligned}}\quad \right|\quad \mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} =\mathbf {A} '\end{matrix}}}

(Q1)

The case n=1 is the w:binary quadratic form introduced by Lagrange (1773) and Gauss (1798/1801), n=2 is the ternary quadratic form introduced by Gauss (1798/1801), n=3 is the quaternary quadratic form etc.

Most general Lorentz transformation

The Lorentz interval is the invariant relation between axes and conjugate diameters of hyperbolas, illustrating Lorentz transformations between two inertial frames.

The general Lorentz transformation follows from (Q1) by setting A=A′=diag(-1,1,...,1) and det g=±1. It forms an w:indefinite orthogonal group called the w:Lorentz group O(1,n), while the case det g=+1 forms the restricted w:Lorentz group SO(1,n). The quadratic form q(x) becomes the w:Lorentz interval in terms of an w:indefinite quadratic form of w:Minkowski space (being a special case of w:pseudo-Euclidean space), and the associated bilinear form b(x) becomes the w:Minkowski inner product:[2][3]

{\displaystyle {\begin{matrix}{\begin{aligned}-x_{0}^{2}+\cdots +x_{n}^{2}&=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}\\-x_{0}y_{0}+\cdots +x_{n}y_{n}&=-x_{0}^{\prime }y_{0}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline \left.{\begin{matrix}\mathbf {x} '=\mathbf {g} \cdot \mathbf {x} \\\downarrow \\{\begin{aligned}x_{0}^{\prime }&=x_{0}g_{00}+x_{1}g_{01}+\dots +x_{n}g_{0n}\\x_{1}^{\prime }&=x_{0}g_{10}+x_{1}g_{11}+\dots +x_{n}g_{1n}\\&\dots \\x_{n}^{\prime }&=x_{0}g_{n0}+x_{1}g_{n1}+\dots +x_{n}g_{nn}\end{aligned}}\\\\\mathbf {x} =\mathbf {g} ^{-1}\cdot \mathbf {x} '\\\downarrow \\{\begin{aligned}x_{0}&=x_{0}^{\prime }g_{00}-x_{1}^{\prime }g_{10}-\dots -x_{n}^{\prime }g_{n0}\\x_{1}&=-x_{0}^{\prime }g_{01}+x_{1}^{\prime }g_{11}+\dots +x_{n}^{\prime }g_{n1}\\&\dots \\x_{n}&=-x_{0}^{\prime }g_{0n}+x_{1}^{\prime }g_{1n}+\dots +x_{n}^{\prime }g_{nn}\end{aligned}}\end{matrix}}\right|{\begin{matrix}{\begin{aligned}\mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }\cdot \mathbf {A} &=\mathbf {g} ^{-1}\\\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} &=\mathbf {A} \\\mathbf {g} \cdot \mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }&=\mathbf {A} \\\\\end{aligned}}\\{\begin{aligned}\sum _{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k}&=\left\{{\begin{aligned}-1\quad &(j=k=0)\\1\quad &(j=k>0)\\0\quad &(j\neq k)\end{aligned}}\right.\\\sum _{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0}&=\left\{{\begin{aligned}-1\quad &(i=k=0)\\1\quad &(i=k>0)\\0\quad &(i\neq k)\end{aligned}}\right.\end{aligned}}\end{matrix}}\end{matrix}}}

(1a)

The invariance of the Lorentz interval with n=1 between axes and w:conjugate diameters of hyperbolas was known for a long time since Apollonius (ca. 200 BC). Lorentz transformations (1a) for various dimensions were used by Gauss (1818), Jacobi (1827, 1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882) in order to simplify computations of w:elliptic functions and integrals.[4][5] They were also used by Chasles (1829) and Weddle (1847) to describe relations on hyperboloids, as well as by Poincaré (1881), Cox (1881-91), Picard (1882, 1884), Killing (1885, 1893), Gérard (1892), Hausdorff (1899), Woods (1901, 1903), Liebmann (1904/05) to describe w:hyperbolic motions (i.e. rigid motions in the w:hyperbolic plane or w:hyperbolic space), which were expressed in terms of Weierstrass coordinates of the w:hyperboloid model satisfying the relation ${\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}=-1}$  or in terms of the w:Cayley–Klein metric of w:projective geometry using the "absolute" form ${\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}=0}$  as discussed by Klein (1871-73).[M 1][6][7] In addition, w:infinitesimal transformations related to the w:Lie algebra of the group of hyperbolic motions were given in terms of Weierstrass coordinates ${\displaystyle -x_{0}^{2}+\cdots +x_{n}^{2}=-1}$  by Killing (1888-1897).

Most general Lorentz transformation of velocity

If ${\displaystyle x_{i},\ x_{i}^{\prime }}$  in (1a) are interpreted as w:homogeneous coordinates, then the corresponding inhomogenous coordinates ${\displaystyle u_{s},\ u_{s}^{\prime }}$  follow by

${\displaystyle {\frac {x_{s}}{x_{0}}}=u_{s},\ {\frac {x_{s}^{\prime }}{x_{0}^{\prime }}}=u_{s}^{\prime }\ (s=1,2\dots n)}$

defined by ${\displaystyle u_{1}^{2}+u_{2}^{2}+\dots +u_{n}^{2}\leq 1}$  so that the Lorentz transformation becomes a w:homography inside the w:unit hypersphere, which w:John Lighton Synge called "the most general formula for the composition of velocities" in terms of special relativity[8] (the transformation matrix g stays the same as in (1a)):

{\displaystyle {\begin{aligned}u_{s}^{\prime }&={\frac {g_{s0}+g_{s1}u_{1}+\dots +g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots +g_{0n}u_{n}}}\\\\u_{s}&={\frac {-g_{0s}+g_{1s}u_{1}^{\prime }+\dots +g_{ns}u_{n}^{\prime }}{g_{00}-g_{10}u_{1}^{\prime }-\dots -g_{n0}u_{n}^{\prime }}}\end{aligned}}\left|{\begin{aligned}\sum _{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k}&=\left\{{\begin{aligned}-1\quad &(j=k=0)\\1\quad &(j=k>0)\\0\quad &(j\neq k)\end{aligned}}\right.\\\sum _{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0}&=\left\{{\begin{aligned}-1\quad &(i=k=0)\\1\quad &(i=k>0)\\0\quad &(i\neq k)\end{aligned}}\right.\end{aligned}}\right.}

(1b)

Such Lorentz transformations for various dimensions were used by Gauss (1818), Jacobi (1827–1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882), Callandreau (1885) in order to simplify computations of elliptic functions and integrals, by Picard (1882-1884) in relation to Hermitian quadratic forms, or by Woods (1901, 1903) in terms of the w:Beltrami–Klein model of hyperbolic geometry. In addition, infinitesimal transformations in terms of the w:Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere ${\displaystyle -1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}=0}$  were given by Lie (1885-1893) and Werner (1889) and Killing (1888-1897).

Historical notation

Apollonius (BC) – Conjugate diameters

Equality of difference in squares

w:Apollonius of Perga (c. 240–190 BC) in his 7th book on conics defined the following well known proposition (the 7th book survived in Arabian translation, and was translated into Latin in 1661 and 1710), as follows:

• In every hyperbola the difference between the squares of the axes is equal to the difference between the squares of any conjugate diameters of the section. (Latin translation 1710 by w:Edmond Halley.)[M 3]
• [..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. (English translation 1896 by w:Thomas Heath.)[M 4]

w:Philippe de La Hire (1685) stated this proposition as follows:

I say that the difference of the squares of any two diameters conjugated to each other, AB, DE, is equal to the difference of the squares of any two other diameters conjugated to each other, NM, LK.[M 5]

and also summarized the related propositions in the 7th book of Apollonius:

In a hyperbola, the difference of the squares of the axes is equal to the difference of the squares of any two conjugate diameters.[M 6]

w:Guillaume de l'Hôpital (1707), using the methods of w:analytic geometry, demonstrated the same proposition:[M 7]

The difference of the squares of any two conjugate diameters "Mm, Ss" is equal to the difference of the squares of the two axes "Aa, Bb." We are to prove that ${\displaystyle {\overline {CS}}^{2}-{\overline {CM}}^{2}={\overline {CB}}^{2}-{\overline {CA}}^{2}}$ , or ${\displaystyle {\overline {CM}}^{2}-{\overline {CS}}^{2}={\overline {CA}}^{2}-{\overline {CB}}^{2}}$ . (English translation 1723 by w:Edmund Stone.)[M 8]
Apollonius' proposition can be expressed as ${\displaystyle -x_{0}^{\prime 2}+x_{1}^{\prime 2}=-x_{0}^{2}+x_{1}^{2}}$  in agreement with the invariance of the Lorentz interval, so that the Lorentz transformation (1a) "(n=1)" can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.

Equality of areas of parallelograms

Apollonius also gave another well known proposition in his 7th book regarding ellipses as well as conjugate sections of hyperbolas (see also Del Centina & Fiocca[9] for further details on the history of this proposition):

• In the ellipse, and in conjugate sections [the opposite branches of two conjugate hyperbolas] the parallelogram bounded by the axes is equal to the parallelogram bounded by any pair of conjugate diameters, if its angles are equal to the angles the conjugate diameters form at the centre. (English translation by Del Centina & Fiocca[10] based on the Latin translation 1661 by w:Giovanni Alfonso Borelli and w:Abraham Ecchellensis.[M 9])
• If two conjugate diameters are taken in an ellipse, or in the opposite conjugate sections; the parallelogram bounded by them is equal to the rectangle bounded by the axes, provided its angles are equal to those formed at the centre by the conjugate diameters. (English translation by Del Centina & Fiocca[10] based on the Latin translation 1710 by w:Edmond Halley.)[M 10])
• If PP', DD' be two conjugate diameters in an ellipse or in conjugate hyperbolas, and if tangents be drawn at the four extremities forming a parallelogram LL'MM', then the parallelogram LL'MM' = rect. AA'·BB'. (English translation 1896 by w:Thomas Heath.)[M 11]
The graphical representation of Apollonius proposition in Borelli's Fig. 4 is essentially a w:Minkowski diagram, being a graphical representation of the Lorentz transformation. If line AB is the x-axis of an inertial frame S1, then line FG is the x-axis of another inertial frames S2 which together with its parallel lines (such as KL and NM) represent w:relativity of simultaneity. Analogously, if line CD is the time axis of another inertial frame S2, then line HI is the time axis of S2 which together with its parallel lines (such as KN and LM) represent the w:worldlines of objects at different locations. The diagonals KE (or KM) and LE (or LN) lie on the asymptotes which form a light cone. Thus the totality of all parallelograms of equal area and conjugate diameters as constructed by Apollonius, represents the totality of all inertial frames, lines of simultaneity and worldlines within a spacetime area bounded by ${\displaystyle -x_{0}^{2}+x_{1}^{2}={\rm {const}}}$ .

w:Grégoire de Saint-Vincent independently (1647) stated the same proposition:[M 12]

The parallelograms whose opposite sides are tangent to two conjugate hyperbolas at the extremities of two conjugate diameters are equivalent among them. (English translation by Del Centina & Fiocca.[11])

w:Philippe de La Hire (1685), who was aware of both Apollonius 7th book and Saint-Vincent's book, stated this proposition as follows:[M 13]

If a parallelogram FGHI is circumscribed about conjugate sections NA, DL, BM, KE whose sides are parallel to two conjugate diameters ED, BA drawn through their extremities, and with similar method another parallelogram OPQR is drawn through the extremities of other two conjugate diameters, then the parallelograms FGHI, OPQR are equal. (English translation by Del Centina & Fiocca.[12])

and also summarized the related propositions in the 7th book of Apollonius:[M 14]

In conjugate sections and in the ellipse, the parallelogram constructed with the axes, is equal to the parallelogram constructed with any two conjugated diameters, provided the angles are equal to those between the diameters themselves. (English translation by Del Centina & Fiocca.[12])
In Saint-Vincent's Fig. 5 or La Hire's Fig. 6, parallelogram FGHI contains all coordinates related to an inertial frame S3, in particular triangles EGH, EFI (Fig. 5) or CFG, CHI (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EHI, EGF (Fig. 5) or CFI, CGH (Fig. 6) contain space like intervals between events on the negative and positive x-axis. Conversely, parallelogram OPQR contains all coordinates related to another frame S4, in particular triangles EQR, EOP (Fig. 5) or CPQ, COR (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EPR, EOQ (Fig. 5) or COP, CQR (Fig. 6) contain space like intervals between events on the negative and positive x-axis.

Lagrange (1773) – Binary quadratic forms

After the invariance of the sum of squares under linear substitutions was discussed by E:Euler (1771), the general expressions of a w:binary quadratic form and its transformation was formulated by w:Joseph-Louis Lagrange (1773/75) as follows[M 15]

{\displaystyle {\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\\hline {\begin{aligned}y&=Ms+Nx\\z&=ms+nx\end{aligned}}\left|{\begin{matrix}{\begin{aligned}P&=pM^{2}+2qMm+rm^{2}\\Q&=pMN+q(Mn+Nm)+rmn\\R&=pN^{2}+2qNn+rn^{2}\end{aligned}}\\\downarrow \\PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}\end{matrix}}\right.\end{matrix}}}
This is equivalent to (Q1) (n=1). The Lorentz interval ${\displaystyle -x_{0}^{2}+x_{1}^{2}}$  and the Lorentz transformation (1a) (n=1) are a special case of the binary quadratic form by setting (p,q,r)=(P,Q,R)=(1,0,-1).

Gauss (1798–1818)

The theory of binary quadratic forms was considerably expanded by w:Carl Friedrich Gauss (1798, published 1801) in his w:Disquisitiones Arithmeticae. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:[M 16]

{\displaystyle {\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\F'=a'x^{\prime 2}+2b'x'y'+c'y^{\prime 2}=(a',b',c')\\\hline {\begin{aligned}x&=\alpha x'+\beta y'\\y&=\gamma x'+\delta y'\\\\x'&=\delta x-\beta y\\y'&=-\gamma x+\alpha y\end{aligned}}\left|{\begin{matrix}{\begin{aligned}a'&=a\alpha ^{2}+2b\alpha \gamma +c\gamma ^{2}\\b'&=a\alpha \beta +b(\alpha \delta +\beta \gamma )+c\gamma \delta \\c'&=a\beta ^{2}+2b\beta \delta +c\delta ^{2}\end{aligned}}\\\downarrow \\b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha \delta -\beta \gamma )^{2}\end{matrix}}\right.\end{matrix}}}

which is equivalent to (Q1) (n=1). As pointed out by Gauss, F and F′ are called "proper equivalent" if αδ-βγ=1, so that F is contained in F′ as well as F′ is contained in F. In addition, if another form F″ is contained by the same procedure in F′ it is also contained in F and so forth.[M 17]

The Lorentz interval ${\displaystyle -x_{0}^{2}+x_{1}^{2}}$  and the Lorentz transformation (1a) (n=1) are a special case of the binary quadratic form by setting (a,b,c)=(a',b',c')=(1,0,-1).

Gauss (1798/1801)[M 18] also discussed ternary quadratic forms with the general expression

{\displaystyle {\begin{matrix}f=ax^{2}+a'x^{\prime 2}+a''x^{\prime \prime 2}+2bx'x''+2b'xx''+2b''xx'=\left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)\\g=my^{2}+m'y^{\prime 2}+m''y^{\prime \prime 2}+2ny'y''+2n'yy''+2n''yy'=\left({\begin{matrix}m,&m',&m''\\n,&n',&n''\end{matrix}}\right)\\\hline {\begin{aligned}x&=\alpha y+\beta y'+\gamma y''\\x'&=\alpha 'y+\beta 'y'+\gamma 'y''\\x''&=\alpha ''y+\beta ''y'+\gamma ''y''\end{aligned}}\end{matrix}}}

which is equivalent to (Q1) (n=2). Gauss called these forms definite when they have the same sign such as x2+y2+z2, or indefinite in the case of different signs such as x2+y2-z2. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:[M 19]

${\displaystyle \left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)\Rightarrow {\begin{matrix}\left({\begin{matrix}1,&-1,&1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&1,&1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}1,&1,&-1\\0,&0,&0\end{matrix}}\right),\\\left({\begin{matrix}1,&-1,&-1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&1,&-1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&-1,&1\\0,&0,&0\end{matrix}}\right)\end{matrix}}}$

These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval ${\displaystyle x^{2}+x^{\prime 2}-x^{\prime \prime 2}}$  and the Lorentz transformation (1a) (n=2) is an indefinite ternary quadratic form, which follows from the general ternary form by setting:

${\displaystyle \left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)=\left({\begin{matrix}m,&m',&m''\\n,&n',&n''\end{matrix}}\right)=\left({\begin{matrix}1,&1,&-1\\0,&0,&0\end{matrix}}\right)}$

Homogeneous coordinates

Gauss (1818) discussed planetary motions together with formulating w:elliptic functions. In order to simplify the integration, he transformed the expression

${\displaystyle (AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E}$

into

${\displaystyle G+G'\cos T^{2}+G''\sin T^{2}}$

in which the w:eccentric anomaly E is connected to the new variable T by the following transformation including an arbitrary constant k, which Gauss then rewrote by setting k=1:[M 20]

{\displaystyle {\begin{matrix}{\scriptstyle \left(\alpha +\alpha '\cos T+\alpha ''\sin T\right)^{2}+\left(\beta +\beta '\cos T+\beta ''\sin T\right)^{2}-\left(\gamma +\gamma '\cos T+\gamma ''\sin T\right)^{2}}=0\\k\left(\cos ^{2}T+\sin ^{2}T-1\right)=0\\\hline {\begin{aligned}\cos E&={\frac {\alpha +\alpha '\cos T+\alpha ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\\\sin E&={\frac {\beta +\beta '\cos T+\beta ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\end{aligned}}\left|{\scriptstyle {\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=k&\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''&=-k\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-k&\beta \beta -\beta '\beta '-\beta ''\beta ''&=-k\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-k&\gamma \gamma -\gamma '\gamma '-\gamma ''\gamma ''&=+k\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0&\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0&\gamma \alpha -\gamma '\alpha '-\gamma ''\alpha ''&=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0&\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\end{aligned}}}\right.\\\hline k=1\\{\begin{aligned}t\cos E&=\alpha +\alpha '\cos T+\alpha ''\sin T\\t\sin E&=\beta +\beta '\cos T+\beta ''\sin T\\t&=\gamma +\gamma '\cos T+\gamma ''\sin T\end{aligned}}\left|{\scriptstyle {\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=1\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-1\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-1\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0\end{aligned}}}\right.\end{matrix}}}

The coefficients α,β,γ,... of Gauss' case k=1 are equivalent to the coefficient system in Lorentz transformations (1a) and (1b) (n=2).

Further setting ${\displaystyle [\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime },\ u_{2}^{\prime }\right]}$ , Gauss' transformation becomes Lorentz transformation (1b) (n=2).

Subsequently, he showed that these relations can be reformulated using three variables x,y,z and u,u′,u″, so that

${\displaystyle aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz}$

can be transformed into

${\displaystyle Guu+G'u'u'+G''u''u''}$ ,

in which x,y,z and u,u′,u″ are related by the transformation:[M 21]

{\displaystyle {\begin{aligned}x&=\alpha u+\alpha 'u'+\alpha ''u''\\y&=\beta u+\beta 'u'+\beta ''u''\\z&=\gamma u+\gamma 'u'+\gamma ''u''\\\\u&=-\alpha x-\beta y+\gamma z\\u'&=\alpha 'x+\beta 'y-\gamma 'z\\u''&=\alpha ''x+\beta ''y-\gamma ''z\end{aligned}}\left|{\scriptstyle {\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=1\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-1\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-1\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0\end{aligned}}}\right.}
This is equivalent to Lorentz transformation (1a) (n=2) satisfying ${\displaystyle x^{2}+y^{2}-z^{2}=u^{\prime 2}+u^{\prime \prime 2}-u^{2}}$ , and can be related to Gauss' previous equations in terms of homogeneous coordinates ${\displaystyle \left[\cos T,\sin T,\cos E,\sin E\right]=\left[{\tfrac {x}{z}},\ {\tfrac {y}{z}},\ {\tfrac {u'}{u}},\ {\tfrac {u''}{u}}\right]}$ .

Jacobi (1827, 1833/34) – Homogeneous coordinates

Following Gauss (1818), w:Carl Gustav Jacob Jacobi extended Gauss' transformation in 1827:[M 22]

{\displaystyle {\scriptstyle {\begin{matrix}\cos P^{2}+\sin P^{2}\cos \vartheta ^{2}+\sin P^{2}\sin \vartheta ^{2}=1\\k\left(\cos \psi ^{2}+\sin \psi ^{2}\cos \varphi ^{2}+\sin \psi ^{2}\sin \varphi ^{2}-1\right)=0\\\hline {\left.{\begin{matrix}\mathbf {(1)} {\begin{aligned}\cos P&={\frac {\alpha +\alpha '\cos \psi +\alpha ''\sin \psi \cos \varphi +\alpha '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\sin P\cos \vartheta &={\frac {\beta +\beta '\cos \psi +\beta ''\sin \psi \cos \varphi +\beta '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\sin P\sin \vartheta &={\frac {\gamma +\beta '\cos \psi +\gamma ''\sin \psi \cos \varphi +\gamma '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\\\cos \psi &={\frac {-\delta '+\alpha '\cos P+\beta '\sin P\cos \vartheta +\gamma '\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\\\sin \psi \cos \varphi &={\frac {-\delta ''+\alpha ''\cos P+\beta ''\sin P\cos \vartheta +\gamma ''\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\\\sin \psi \sin \varphi &={\frac {-\delta '''+\alpha '''\cos P+\beta '''\sin P\cos \vartheta +\gamma '''\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\end{aligned}}\\\\\hline \mathbf {(2)} {\begin{aligned}\alpha \mu +\beta x+\gamma y+\delta z&=m\\\alpha '\mu +\beta 'x+\gamma 'y+\delta 'z&=m'\\\alpha ''\mu +\beta ''x+\gamma ''y+\delta ''z&=m''\\\alpha '''\mu +\beta '''x+\gamma '''y+\delta '''z&=m'''\\\\Am+A'm'+A''m''+A'''m'''&=\mu \\Bm+B'm'+B''m''+B'''m'''&=x\\Cm+C'm'+C''m''+C'''m'''&=y\\Dm+D'm'+D''m''+D'''m'''&=z\\\\\end{aligned}}\\{\begin{aligned}\alpha &=-kA,&\beta &=-kB,&\gamma &=-kC,&\delta &=kD,\\\alpha '&=kA',&\beta '&=kB',&\gamma '&=kC',&\delta '&=-kD',\\\alpha ''&=kA'',&\beta ''&=kB'',&\gamma ''&=kC'',&\delta ''&=-kD'',\\\alpha '''&=kA''',&\beta '''&=kB''',&\gamma '''&=kC''',&\delta '''&=-kD''',\end{aligned}}\end{matrix}}\right|{\begin{matrix}{\begin{aligned}\alpha \alpha +\beta \beta +\gamma \gamma -\delta \delta &=-k\\\alpha '\alpha '+\beta '\beta '+\gamma '\gamma '-\delta '\delta '&=k\\\alpha ''\alpha ''+\beta ''\beta ''+\gamma ''\gamma ''-\delta ''\delta ''&=k\\\alpha '''\alpha '''+\beta '''\beta '''+\gamma '''\gamma '''-\delta '''\delta '''&=k\\\alpha \alpha '+\beta \beta '+\gamma \gamma '-\delta \delta '&=0\\\alpha \alpha ''+\beta \beta ''+\gamma \gamma ''-\delta \delta ''&=0\\\alpha \alpha '''+\beta \beta '''+\gamma \gamma '''-\delta \delta '''&=0\\\alpha ''\alpha '''+\beta ''\beta '''+\gamma ''\gamma '''-\delta ''\delta '''&=0\\\alpha '''\alpha '+\beta '''\beta '+\gamma '''\gamma '-\delta '''\delta '&=0\\\alpha '\alpha ''+\beta '\beta ''+\gamma '\gamma ''-\delta '\delta ''&=0\\\\-\alpha \alpha +\alpha '\alpha '+\alpha ''\alpha ''+\alpha '''\alpha '''&=k\\-\beta \beta +\beta '\beta '+\beta ''\beta ''+\beta '''\beta '''&=k\\-\gamma \gamma +\gamma '\gamma '+\gamma ''\gamma ''+\gamma '''\gamma '''&=k\\-\delta \delta +\delta '\delta '+\delta ''\delta ''+\delta '''\delta '''&=-k\\-\alpha \beta +\alpha '\beta '+\alpha ''\beta ''+\alpha '''\beta '''&=0\\-\alpha \gamma +\alpha '\gamma '+\alpha ''\gamma ''+\alpha '''\gamma '''&=0\\-\alpha \delta +\alpha '\delta '+\alpha ''\delta ''+\alpha '''\delta '''&=0\\-\beta \gamma +\beta '\gamma '+\beta ''\gamma ''+\beta '''\gamma '''&=0\\-\gamma \delta +\gamma '\delta '+\gamma ''\delta ''+\gamma '''\delta '''&=0\\-\delta \beta +\delta '\beta '+\delta ''\beta ''+\delta '''\beta '''&=0\end{aligned}}\end{matrix}}}\end{matrix}}}}
By setting {\displaystyle {\scriptstyle {\begin{aligned}\left[\cos P,\ \sin P\cos \varphi ,\ \sin P\sin \varphi \right]&=\left[u_{1},\ u_{2},\ u_{3}\right]\\{}[\cos \psi ,\ \sin \psi \cos \vartheta ,\ \sin \psi \sin \vartheta ]&=\left[u_{1}^{\prime },\ u_{2}^{\prime },\ u_{3}^{\prime }\right]\end{aligned}}}}  and k=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation (1b) (n=3), and by setting k=1 in transformation system (2) it becomes equivalent to Lorentz transformation (1a) (n=3) producing ${\displaystyle m^{2}+m^{\prime 2}+m^{\prime \prime 2}-m^{\prime \prime \prime 2}=\mu ^{2}+x^{2}+y^{2}-z^{2}}$ .

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):[M 23]

{\displaystyle {\scriptstyle {\begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\\mathbf {(1)} {\begin{aligned}x&=\alpha s+\alpha 's'+\alpha ''s''\\y&=\beta s+\beta 's'+\beta ''s''\\z&=\gamma s+\gamma 's'+\gamma ''s''\\\\s&=\alpha x+\beta y+\gamma z\\s'&=\alpha 'x+\beta 'y+\gamma 'z\\s''&=\alpha ''x+\beta ''y+\gamma ''z\end{aligned}}\left|{\begin{aligned}\alpha \alpha +\beta \beta +\gamma \gamma &=1&\alpha \alpha +\alpha '\alpha '+\alpha ''\alpha ''&=1\\\alpha '\alpha '+\beta '\beta '+\gamma '\gamma '&=1&\beta \beta +\beta '\beta '+\beta ''\beta ''&=1\\\alpha ''\alpha ''+\beta ''\beta ''+\gamma ''\gamma ''&=1&\gamma \gamma +\gamma '\gamma '+\gamma ''\gamma ''&=1\\\alpha '\alpha ''+\beta '\beta ''+\gamma '\gamma ''&=0&\beta \gamma +\beta '\gamma '+\beta ''\gamma ''&=0\\\alpha ''\alpha +\beta ''\beta +\gamma ''\gamma &=0&\gamma \alpha +\gamma '\alpha '+\gamma ''\alpha ''&=0\\\alpha \alpha '+\beta \beta '+\gamma \gamma '&=0&\alpha \beta +\alpha '\beta '+\alpha ''\beta ''&=0\end{aligned}}\right.\\\hline \left[{\frac {y}{x}},\ {\frac {z}{x}},\ {\frac {s'}{s}},\ {\frac {s''}{s}}\right]=\left[-i\cos \varphi ,\ -i\sin \varphi ,\ i\cos \eta ,\ i\sin \eta \right]\\\left[\alpha ',\ \alpha '',\ \beta ,\ \gamma \right]=\left[i\alpha ',\ i\alpha '',\ -i\beta ,\ -i\gamma \right]\\\hline {\begin{matrix}\mathbf {(2)} {\begin{matrix}\left(\alpha -\alpha '\cos \eta -\alpha ''\sin \eta \right)^{2}=\left(\beta -\beta '\cos \eta -\beta ''\sin \eta \right)^{2}+\left(\gamma -\gamma '\cos \eta -\gamma ''\sin \eta \right)^{2}\\\left(\alpha -\beta \cos \phi -\gamma \sin \phi \right)^{2}=\left(\alpha '-\beta '\cos \phi -\gamma '\sin \phi \right)^{2}+\left(\alpha ''-\beta ''\cos \phi -\gamma ''\sin \phi \right)^{2}\\\hline {\begin{aligned}\cos \phi &={\frac {\beta -\beta '\cos \eta -\beta ''\sin \eta }{\alpha -\alpha '\cos \eta -\alpha ''\sin \eta }},&\cos \eta &={\frac {\alpha '-\beta '\cos \phi -\gamma '\sin \phi }{\alpha -\beta \cos \phi -\gamma \sin \phi }}\\\sin \phi &={\frac {\gamma -\gamma '\cos \eta -\gamma ''\sin \eta }{\alpha -\alpha '\cos \eta -\alpha ''\sin \eta }},&\sin \eta &={\frac {\alpha ''-\beta ''\cos \phi -\gamma ''\sin \phi }{\alpha -\beta \cos \phi -\gamma \sin \phi }}\end{aligned}}\end{matrix}}\\\hline \\\mathbf {(3)} {\begin{matrix}1-zz-yy={\frac {1-s's'-s''s''}{\left(\alpha -\alpha 's'-\alpha ''s''\right)^{2}}}\\\hline {\begin{aligned}y&={\frac {\beta -\beta 's'-\beta ''s''}{\alpha -\alpha 's'-\alpha ''s''}},&s'&={\frac {\alpha '-\beta 'y-\gamma 'z}{\alpha -\beta y-\gamma z}},\\z&={\frac {\gamma -\gamma 's'-\gamma ''s''}{\alpha -\alpha 's'-\alpha ''s'''}},&s''&={\frac {\alpha ''-\beta ''y-\gamma ''z}{\alpha -\beta y-\gamma z}},\end{aligned}}\end{matrix}}\end{matrix}}\left|{\begin{aligned}\alpha \alpha -\beta \beta -\gamma \gamma &=1\\\alpha '\alpha '-\beta '\beta '-\gamma '\gamma '&=-1\\\alpha ''\alpha ''-\beta ''\beta ''-\gamma ''\gamma ''&=-1\\\alpha '\alpha ''-\beta '\beta ''-\gamma '\gamma ''&=0\\\alpha ''\alpha -\beta ''\beta -\gamma ''\gamma &=0\\\alpha \alpha '-\beta \beta '-\gamma \gamma '&=0\\\\\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''&=1\\\beta \beta -\beta '\beta '-\beta ''\beta ''&=-1\\\gamma \gamma -\gamma '\gamma '-\gamma ''\gamma ''&=-1\\\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\\\gamma \alpha -\gamma '\alpha '-\gamma ''\alpha ''&=0\\\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\end{aligned}}\right.\end{matrix}}}}
By setting ${\displaystyle [\cos \phi ,\ \sin \phi ,\ \cos \eta ,\ \sin \eta ]=\left[u_{1},\ u_{2},\ u_{1}^{\prime },\ u_{2}^{\prime }\right]}$ , transformation system (2) is equivalent to Lorentz transformation (1b) (n=2). Also transformation system (3) is equivalent to Lorentz transformation (1b) (n=2) up to a sign change.

Extending his previous result, Jacobi (1833) started with Cauchy's (1829) orthogonal transformation for n dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of n dimensions:[M 24]

{\displaystyle {\scriptstyle {\begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots +x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots +y_{n}y_{n}\\\hline \mathbf {(1)\ } {\begin{aligned}y_{\varkappa }&=\alpha _{1}^{(\varkappa )}x_{1}+\alpha _{2}^{(\varkappa )}x_{2}+\dots +\alpha _{n}^{(\varkappa )}x_{n}\\x_{\varkappa }&=\alpha _{\varkappa }^{\prime }y_{1}+\alpha _{\varkappa }^{\prime \prime }y_{2}+\dots +\alpha _{\varkappa }^{(n)}y_{n}\\\\{\frac {y_{\varkappa }}{y_{n}}}&={\frac {\alpha _{1}^{(\varkappa )}x_{1}+\alpha _{2}^{(\varkappa )}x_{2}+\dots +\alpha _{n}^{(\varkappa )}x_{n}}{\alpha _{1}^{(n)}x_{1}+\alpha _{2}^{(n)}x_{2}+\dots +\alpha _{n}^{(n)}x_{n}}}\\{\frac {x_{\varkappa }}{x_{n}}}&={\frac {\alpha _{\varkappa }^{\prime }y_{1}+\alpha _{\varkappa }^{\prime \prime }y_{2}+\dots +\alpha _{\varkappa }^{(n)}y_{n}}{\alpha _{1}^{(n)}x_{1}+\alpha _{2}^{(n)}x_{2}+\dots +\alpha _{n}^{(n)}x_{n}}}\end{aligned}}\left|{\begin{aligned}\alpha _{\varkappa }^{\prime }\alpha _{\lambda }^{\prime }+\alpha _{\varkappa }^{\prime \prime }\alpha _{\lambda }^{\prime \prime }+\dots +\alpha _{\varkappa }^{(n)}\alpha _{\lambda }^{(n)}&=0\\\alpha _{\varkappa }^{\prime }\alpha _{\varkappa }^{\prime }+\alpha _{\varkappa }^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }+\dots +\alpha _{\varkappa }^{(n)}\alpha _{\varkappa }^{(n)}&=1\\\\\alpha _{1}^{(\varkappa )}\alpha _{1}^{(\lambda )}+\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\lambda )}+\dots +\alpha _{n}^{(\varkappa )}\alpha _{n}^{(\lambda )}&=0\\\alpha _{1}^{(\varkappa )}\alpha _{1}^{(\varkappa )}+\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\varkappa )}+\dots +\alpha _{n}^{(\varkappa )}\alpha _{n}^{(\varkappa )}&=1\end{aligned}}\right.\\\hline {\frac {x_{\varkappa }}{x_{n}}}=-i\xi _{\varkappa },\ {\frac {y_{\varkappa }}{y_{n}}}=i\nu _{\varkappa }\\1-\xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}={\frac {y_{n}y_{n}}{x_{n}x_{n}}}\left(1-\nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}\right)\\\alpha _{n}^{(\varkappa )}=i\alpha ^{(\varkappa )},\ \alpha _{\varkappa }^{(n)}=-i\alpha _{\varkappa },\ \alpha _{n}^{(n)}=\alpha \\1-\xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}={\frac {1-\nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}}{\left[\alpha -\alpha ^{\prime }\nu _{1}-\alpha ^{\prime \prime }\nu _{2}\dots -\alpha ^{(n-1)}\nu _{n-1}\right]^{2}}}\\\hline \mathbf {(2)\ } {\begin{aligned}\nu _{\varkappa }&={\frac {\alpha ^{(\varkappa )}-\alpha _{1}^{(\varkappa )}\xi _{1}-\alpha _{2}^{(\varkappa )}\xi _{2}\dots -\alpha _{n-1}^{(\varkappa )}\xi _{n-1}}{\alpha -\alpha _{1}\xi _{1}-\alpha _{2}\xi _{2}\dots -\alpha _{n-1}\xi _{n-1}}}\\\\\xi _{\varkappa }&={\frac {\alpha _{\varkappa }-\alpha _{\varkappa }^{\prime }\nu _{1}-\alpha _{2}^{\prime \prime }\nu _{2}\dots -\alpha _{\varkappa }^{(n-1)}\nu _{n-1}}{\alpha -\alpha ^{\prime }\nu _{1}-\alpha ^{\prime \prime }\nu _{2}\dots -\alpha ^{(n-1)}\nu _{n-1}}}\end{aligned}}\\\hline \xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}=1\ \Rightarrow \ \nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}=1\end{matrix}}}}
Transformation system (2) is equivalent to Lorentz transformation (1b) up to a sign change.

He also stated the following transformation leaving invariant the Lorentz interval:[M 25]

{\displaystyle {\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots -u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots -w_{n-1}w_{n-1}\\\hline {\scriptstyle {\begin{aligned}u&=\alpha w-\alpha 'w_{1}-\alpha ''w_{2}-\dots -\alpha ^{(n-1)}w_{n-1}\\u_{1}&=\alpha _{1}w-\alpha _{1}^{\prime }w_{1}-\alpha _{1}^{\prime \prime }w_{2}-\dots -\alpha _{1}^{(n-1)}w_{n-1}\\&\dots \\u_{n-1}&=\alpha _{n-1}w-\alpha _{n-1}^{\prime }w_{1}-\alpha _{n-1}^{\prime \prime }w_{2}-\dots -\alpha _{n-1}^{(n-1)}w_{n-1}\\\\w&=\alpha u-\alpha _{1}u_{1}-\alpha _{2}^{\prime \prime }u_{2}-\dots -\alpha _{n-1}u_{n-1}\\w_{1}&=\alpha 'u-\alpha _{1}^{\prime }u_{1}-\alpha _{2}^{\prime }u_{2}-\dots -\alpha _{n-1}^{\prime }u_{n-1}\\&\dots \\w_{n-1}&=\alpha ^{(n-1)}u-\alpha _{1}^{(n-1)}u_{1}-\alpha _{2}^{(n-1)}u_{2}-\dots -\alpha _{n-1}^{(n-1)}u_{n-1}\end{aligned}}\left|{\begin{aligned}\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''\dots -\alpha ^{(n-1)}\alpha ^{(n-1)}&=+1\\\alpha _{\varkappa }\alpha _{\varkappa }-\alpha _{\varkappa }^{\prime }\alpha _{\varkappa }^{\prime }-\alpha _{\varkappa }^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }\dots -\alpha _{\varkappa }^{(n-1)}\alpha _{\varkappa }^{(n-1)}&=-1\\\alpha \alpha _{\varkappa }-\alpha ^{\prime }\alpha _{\varkappa }^{\prime }-\alpha ^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }\dots -\alpha ^{(n-1)}\alpha _{\varkappa }^{(n-1)}&=0\\\alpha _{\varkappa }\alpha _{\lambda }-\alpha _{\varkappa }^{\prime }\alpha _{\lambda }^{\prime }-\alpha _{\varkappa }^{\prime \prime }\alpha _{\lambda }^{\prime \prime }\dots -\alpha _{\varkappa }^{(n-1)}\alpha _{\lambda }^{(n-1)}&=0\\\\\alpha \alpha -\alpha _{1}\alpha _{1}-\alpha _{2}\alpha _{2}\dots -\alpha _{n-1}\alpha _{n-1}&=+1\\\alpha _{\varkappa }\alpha _{\varkappa }-\alpha _{1}^{\varkappa }\alpha _{1}^{\varkappa }-\alpha _{2}^{\prime \prime }\alpha _{2}^{\prime \prime }\dots -\alpha _{n-1}^{(\varkappa )}\alpha _{n-1}^{(\varkappa )}&=-1\\\alpha \alpha ^{(\varkappa )}-\alpha _{1}\alpha _{1}^{(\varkappa )}-\alpha _{2}\alpha _{2}^{(\varkappa )}\dots -\alpha _{n-1}\alpha _{n-1}^{(\varkappa )}&=0\\\alpha ^{(\varkappa )}\alpha ^{(\lambda )}-\alpha _{1}^{(\varkappa )}\alpha _{1}^{\lambda l)}-\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\lambda )}\dots -\alpha _{n-1}^{(\varkappa )}\alpha _{n-1}^{(\lambda )}&=0\end{aligned}}{\text{ }}\right.}\end{matrix}}}
This is equivalent to Lorentz transformation (1a) up to a sign change.

Chasles (1829) – Conjugate hyperboloids

w:Michel Chasles (1829) independently introduced the same equation systems as Gauss (1818) and Jacobi (1827), albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:[M 26]

{\displaystyle {\begin{matrix}\left.{\begin{aligned}\alpha ^{2}+\beta ^{2}-\gamma ^{2}&=1\\\alpha ^{\prime 2}+\beta ^{\prime 2}-\gamma ^{\prime 2}&=1\\\alpha ^{\prime \prime 2}+\beta ^{\prime \prime 2}-\gamma ^{\prime \prime 2}&=-1\end{aligned}}\right\}&\dots (a)\\\\\left.{\begin{aligned}\alpha \alpha '+\beta \beta '-\gamma \gamma '&=0\\\alpha \alpha ''+\beta \beta ''-\gamma \gamma ''&=0\\\alpha '\alpha ''+\beta '\beta ''-\gamma '\gamma '&=0\end{aligned}}\right\}&\dots (b)\\\\\left.{\begin{aligned}\alpha ^{2}+\alpha ^{\prime 2}-\alpha ^{\prime \prime 2}&=1\\\beta ^{2}+\beta ^{\prime 2}-\beta ^{\prime \prime 2}&=1\\\gamma ^{2}+\gamma ^{\prime 2}-\gamma ^{\prime \prime 2}&=-1\end{aligned}}\right\}&\dots (c)\\\\\left.{\begin{aligned}\alpha \beta +\alpha '\beta '-\alpha ''\beta ''&=0\\\alpha \gamma +\alpha '\gamma '-\alpha ''\gamma ''&=0\\\beta \gamma +\beta '\gamma '-\beta ''\gamma ''&=0\end{aligned}}\right\}&\dots (d)\end{matrix}}}

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by E:Euler (1771)] by setting:[M 27]

${\displaystyle {\begin{matrix}\gamma \quad \Rightarrow \quad -\gamma {\sqrt {-1}}\\\gamma '\quad \Rightarrow \quad -\gamma '{\sqrt {-1}}\\\alpha ''\quad \Rightarrow \quad \alpha ''{\sqrt {-1}}\\\beta ''\quad \Rightarrow \quad \beta ''{\sqrt {-1}}\end{matrix}}}$
Equations (a,b,c,d) are the coefficients of Lorentz transformation (1a, n=2).

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone ${\displaystyle {\tfrac {x^{2}}{a{{}^{2}}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=0}$ . He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:[M 28]

{\displaystyle {\begin{matrix}{\frac {x^{2}}{a{{}^{2}}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1\\{\frac {x^{2}}{a{{}^{2}}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\\\hline {\begin{aligned}x&=lx'+l'y'+l''z'\\y&=mx'+m'y'+m''z'\\z&=nx'+n'y'+n''z'\end{aligned}}\\\left\{{\begin{aligned}{\frac {ll'}{a{{}^{2}}}}+{\frac {mm'}{b^{2}}}-{\frac {nn'}{c^{2}}}&=0\\{\frac {ll''}{a{{}^{2}}}}+{\frac {mm''}{b^{2}}}-{\frac {nn''}{c^{2}}}&=0\\{\frac {l'l''}{a{{}^{2}}}}+{\frac {m'm''}{b^{2}}}-{\frac {n'n''}{c^{2}}}&=0\end{aligned}}\right\}\\\hline \left({\frac {l^{2}}{a{{}^{2}}}}+{\frac {m^{2}}{b^{2}}}-{\frac {n^{2}}{c^{2}}}\right)x^{\prime 2}+\left({\frac {l^{\prime 2}}{a{{}^{2}}}}+{\frac {m^{\prime 2}}{b^{2}}}-{\frac {n^{\prime 2}}{c^{2}}}\right)y^{\prime 2}+\left({\frac {l^{\prime \prime 2}}{a{{}^{2}}}}+{\frac {m^{\prime \prime 2}}{b^{2}}}-{\frac {n^{\prime \prime 2}}{c^{2}}}\right)z^{\prime 2}=1\\\left({\frac {l^{2}}{a{{}^{2}}}}+{\frac {m^{2}}{b^{2}}}-{\frac {n^{2}}{c^{2}}}\right)x^{\prime 2}+\left({\frac {l^{\prime 2}}{a{{}^{2}}}}+{\frac {m^{\prime 2}}{b^{2}}}-{\frac {n^{\prime 2}}{c^{2}}}\right)y^{\prime 2}+\left({\frac {l^{\prime \prime 2}}{a{{}^{2}}}}+{\frac {m^{\prime \prime 2}}{b^{2}}}-{\frac {n^{\prime \prime 2}}{c^{2}}}\right)z^{\prime 2}=-1\end{matrix}}}
Chasles defined the conditional equations of l,m,n in the same way as those of ${\displaystyle \alpha ,\beta ,\gamma }$  in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation (1a, n=2) by applying equation system (a) as well.

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids”:[M 29]

{\displaystyle {\begin{matrix}\left.{\begin{aligned}\alpha ^{2}+\beta ^{2}-\gamma ^{2}&=a^{2}\\\alpha ^{\prime 2}+\beta ^{\prime 2}-\gamma ^{\prime 2}&=b^{2}\\\alpha ^{\prime \prime 2}+\beta ^{\prime \prime 2}-\gamma ^{\prime \prime 2}&=-c^{2}\end{aligned}}\right\}&\dots (A)\\\left.{\begin{aligned}\alpha \alpha '+\beta \beta '-\gamma \gamma '&=0\\\alpha \alpha ''+\beta \beta ''-\gamma \gamma ''&=0\\\alpha '\alpha ''+\beta '\beta ''-\gamma '\gamma '&=0\end{aligned}}\right\}&\dots (B)\\\left.{\begin{aligned}\alpha ^{2}+\alpha ^{\prime 2}-\alpha ^{\prime \prime 2}&=a^{2}\\\beta ^{2}+\beta ^{\prime 2}-\beta ^{\prime \prime 2}&=b^{2}\\\gamma ^{2}+\gamma ^{\prime 2}-\gamma ^{\prime \prime 2}&=-c^{2}\end{aligned}}\right\}&\dots (C)\\\left.{\begin{aligned}\alpha \beta +\alpha '\beta '-\alpha ''\beta ''&=0\\\alpha \gamma +\alpha '\gamma '-\alpha ''\gamma ''&=0\\\beta \gamma +\beta '\gamma '-\beta ''\gamma ''&=0\end{aligned}}\right\}&\dots (D)\end{matrix}}}
Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation (1a, n=2) by setting a=b=c=1.

Lebesgue (1837) – Homogeneous coordinates

w:Victor-Amédée Lebesgue (1837) summarized the previous work of Gauss (1818), Jacobi (1827, 1833), Cauchy (1829). He started with the orthogonal transformation[M 30]

{\displaystyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}\ (9)\\\hline {\scriptstyle {\begin{aligned}x_{1}&=a_{1,1}y_{1}+a_{1,2}y_{2}+\dots +a_{1,n}y_{n}\\x_{2}&=a_{2,1}y_{1}+a_{2,2}y_{2}+\dots +a_{2,n}y_{n}\\\dots \\x_{n}&=a_{n,1}x_{1}+a_{n,2}x_{2}+\dots +a_{n,n}x_{n}\\\\y_{1}&=a_{1,1}x_{1}+a_{2,1}x_{2}+\dots +a_{n,1}x_{n}\\y_{2}&=a_{1,2}x_{1}+a_{2,2}x_{2}+\dots +a_{n,2}x_{n}\ (12)\ \\\dots \\y_{n}&=a_{1,n}x_{1}+a_{2,n}x_{2}+\dots +a_{n,n}x_{n}\end{aligned}}\left|{\begin{aligned}a_{1,\alpha }^{2}+a_{2,\alpha }^{2}+\dots +a_{n,\alpha }^{2}&=1&(10)\\a_{1,\alpha }a_{1,\beta }+a_{2,\alpha }a_{2,\beta }+\dots +a_{n,\alpha }a_{n,\beta }&=0&(11)\\a_{\alpha ,1}^{2}+a_{\alpha ,2}^{2}+\dots +a_{\alpha ,n}^{2}&=1&(13)\\a_{\alpha ,1}a_{\beta ,1}+a_{\alpha ,2}a_{\beta ,2}+\dots +a_{\alpha ,n}a_{\beta ,n}&=0&(14)\end{aligned}}\right.}\end{matrix}}}

In order to achieve the invariance of the Lorentz interval[M 31]

${\displaystyle x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}}$

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first n-1 equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=n change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the n-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the n-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

These instructions give Lorentz transformation (1a) in the form:

{\displaystyle {\scriptstyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}\\\hline {\begin{aligned}x_{1}&=a_{1,1}y_{1}+a_{1,2}y_{2}+\dots +a_{1,n}y_{n}\\x_{2}&=a_{2,1}y_{1}+a_{2,2}y_{2}+\dots +a_{2,n}y_{n}\\\dots \\x_{n}&=a_{n,1}x_{1}+a_{n,2}x_{2}+\dots +a_{n,n}x_{n}\\\\y_{1}&=a_{1,1}x_{1}+a_{2,1}x_{2}+\dots +a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\y_{2}&=a_{1,2}x_{1}+a_{2,2}x_{2}+\dots +a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\\dots \\-y_{n}&=a_{1,n}x_{1}+a_{2,n}x_{2}+\dots +a_{n-1,n}x_{n-1}-a_{n,n}x_{n}\end{aligned}}\left|{\begin{aligned}a_{1,\alpha }^{2}+a_{2,\alpha }^{2}+\dots +a_{n-1,\alpha }^{2}-a_{n,\alpha }^{2}&=1\\a_{1,n}^{2}+a_{2,n}^{2}+\dots +a_{n-1,n}^{2}-a_{n,n}^{2}&=-1\\a_{1,\alpha }a_{1,\beta }+a_{2,\alpha }a_{2,\beta }+\dots +a_{n-1,\alpha }a_{n-1,\beta }-a_{n,\alpha }a_{n,\beta }&=0\\a_{\alpha ,1}^{2}+a_{\alpha ,2}^{2}+\dots +a_{\alpha ,n-1}^{2}-a_{\alpha ,n}^{2}&=1\\a_{n,1}^{2}+a_{n,2}^{2}+\dots +a_{n,n-1}^{2}-a_{n,n}^{2}&=-1\\a_{\alpha ,1}a_{\beta ,1}+a_{\alpha ,2}a_{\beta ,2}+\dots +a_{\alpha ,n-1}a_{\beta ,n-1}-a_{\alpha ,n}a_{\beta ,n}&=0\end{aligned}}\right.\end{matrix}}}}

He went on to redefine the variables of the Lorentz interval and its transformation:[M 32]

{\displaystyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}\\\downarrow \\{\begin{aligned}x_{1}&=x_{n}\cos \theta _{1},&x_{2}&=x_{n}\cos \theta _{2},\dots &x_{n-1}&=x_{n}\cos \theta _{n-1}\\y_{1}&=y_{n}\cos \phi _{1},&y_{2}&=y_{n}\cos \phi _{2},\dots &y_{n-1}&=y_{n}\cos \phi _{n-1}\end{aligned}}\\\downarrow \\\cos ^{2}\theta _{1}+\cos ^{2}\theta _{2}+\dots +\cos ^{2}\theta _{n-1}=1\\\cos ^{2}\phi _{1}+\cos ^{2}\phi _{2}+\dots +\cos ^{2}\phi _{n-1}=1\\\hline \\\cos \theta _{i}={\frac {a_{i,1}\cos \phi _{1}+a_{i,2}\cos \phi _{2}+\dots +a_{i,n-1}\cos \phi _{n-1}+a_{i,n}}{a_{n,1}\cos \phi _{1}+a_{n,2}\cos \phi _{2}+\dots +a_{n,n-1}\cos \phi _{n-1}+a_{n,n}}}\\(i=1,2,3\dots n)\end{matrix}}}
Setting ${\displaystyle [\cos \theta _{i},\ \cos \phi _{i}]=\left[u_{s},\ u_{s}^{\prime }\right]}$  it is equivalent to Lorentz transformation (1b).

Weddle (1847) – Conjugate hyperboloids

Very similar to Chasles (1829), though without reference to him, w:Thomas Weddle discussed conjugate hyperboloids using the following equation system (α), from which he derived equations (β) and others:[M 33]

{\displaystyle {\begin{matrix}\left.{\begin{aligned}l_{1}^{2}+m_{1}^{2}-n_{1}^{2}&=1,&l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2}&=0\\l_{2}^{2}+m_{2}^{2}-n_{2}^{2}&=1,&l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3}&=0\\l_{3}^{2}+m_{3}^{2}-n_{3}^{2}&=-1,&l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3}&=0\end{aligned}}\right\}&\dots (\alpha )\\\\\left.{\begin{aligned}l_{1}^{2}+l_{2}^{2}-l_{3}^{2}&=1,&l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3}&=0\\m_{1}^{2}+m_{2}^{2}-m_{3}^{2}&=1,&l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3}&=0\\n_{1}^{2}+n_{2}^{2}-n_{3}^{2}&=-1,&m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3}&=0\end{aligned}}\right\}&\dots (\beta )\end{matrix}}}
These are the coefficients of Lorentz transformation (1a, n=2).

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone ${\displaystyle {\tfrac {x^{2}}{a{{}^{2}}}}+{\tfrac {y^{2}}{b^{2}}}-{\tfrac {z^{2}}{c^{2}}}=0}$ , he defined three conjugate points ${\displaystyle (x_{1}\dots ,y_{1}\dots ,z_{1}\dots )}$  on those two conjugate hyperboloids, related to each other in the same way as equations (α, β) stated above:[M 34]

{\displaystyle {\begin{matrix}{\frac {x^{2}}{a{{}^{2}}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1\\{\frac {x^{2}}{a{{}^{2}}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1\\\hline {\begin{aligned}{\frac {x_{1}x_{2}}{a{{}^{2}}}}+{\frac {y_{1}y_{2}}{b^{2}}}-{\frac {z_{1}z_{2}}{c^{2}}}&=0\\{\frac {x_{1}x_{3}}{a{{}^{2}}}}+{\frac {y_{1}y_{3}}{b^{2}}}-{\frac {z_{1}z_{3}}{c^{2}}}&=0\\{\frac {x_{2}x_{3}}{a{{}^{2}}}}+{\frac {y_{2}y_{3}}{b^{2}}}-{\frac {z_{2}z_{3}}{c^{2}}}&=0\end{aligned}}\quad {\begin{aligned}{\frac {x_{1}^{2}}{a{{}^{2}}}}+{\frac {y_{1}^{2}}{b^{2}}}-{\frac {z_{1}^{2}}{c^{2}}}&=1\\{\frac {x_{2}^{2}}{a{{}^{2}}}}+{\frac {y_{2}^{2}}{b^{2}}}-{\frac {z_{2}^{2}}{c^{2}}}&=1\\{\frac {x_{3}^{2}}{a{{}^{2}}}}+{\frac {y_{3}^{2}}{b^{2}}}-{\frac {z_{3}^{2}}{c^{2}}}&=-1\end{aligned}}\\{\begin{aligned}x_{1}^{2}+x_{2}^{2}-x_{3}^{2}&=a^{2}\\y_{1}^{2}+y_{2}^{2}-y_{3}^{2}&=b^{2}\\z_{1}^{2}+z_{2}^{2}-z_{3}^{2}&=-c^{2}\end{aligned}}\quad {\begin{aligned}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3}&=0\\x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3}&=0\\y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3}&=0\end{aligned}}\end{matrix}}}
These are the coefficients of Lorentz transformation (1a, n=2) by setting a=b=c=1.

Bour (1856) – Homogeneous coordinates

Following Gauss (1818), w:Edmond Bour (1856) wrote the transformations:[M 35]