W 
then putting x 2 =-2% — x x , &c., the proposed equation will be satisfied if only 
£ 2 + »? 2 + £"+ <» 2 = + Wi + K z \ + ww i> 
which will obviously be the case if 
x 1 = £+ vrj — /x£+ aw, 
y X = — Vg + 7) + \£+ba), 
z x = fl% — Xr] + £ 4- eta, 
w x = — ag — brj — c£ 4- a), 
where \, /a, r, a, b, c are arbitrary. 
Write for shortness 
aX + bfx 4- cv — (f>, 1 4- X 2 + ¡i? 4- y 2 + a 2 4- b 2 + c 2 4- cf> 2 = &, 
then we have 
&£ = (1 + X 2 4- 6 2 4- c 2 ) x x + (X/x — v — oh — c4>) y x + (vX + ¡x — ca + bcf>) ^ + (&z> -cp-a — Xcf>)7V x , 
&?7 =(X/u, 4- v — ab+ccf)) x x 4- (1 4- p 2 + c 2 + a 2 ) ^ + (/¿z/ — \ — be + a<f>) z x 4- (cX -av — b- pcf>)w x , 
Jc£ = (vX — /jl — ca— bcf>)x 1 + (/xv + X — be + acf>)y x + (1 +v 2 + a 2 +b 2 )z x + (ap—bX — c — vcf>)w 1 , 
kw = (bv — cp+ a +Xcf))x x + (cX —av+b +pcf))y x + (ap—bv+ c+vcf>)z 1 + (l + A, 2 4-/a 2 4- v 2 ) w x ; 
and from these we obtain 
kx 2 = (1 + X 2 + b 2 + c 2 — p 2 — v 2 — a 2 — </> 2 ) x x 4- 2 (Xp — v — ab — ccf)) y x + 2 (i/A, + p — ca + &<£) ^ 
4- 2 (bv — CfJL — a — Xcf>) w x , 
ky 2 = 2 (Xp + v — ab + ccf)) #! + (1 + /jl 2 + c 2 +a 2 —v 2 — X 2 — b 2 — efr) y x + 2 (pv — X—bc — acf>) z x 
+ 2 (cX — av — b — pcf>) w x , 
kz 2 = 2 (vX — p — ca — bcf>) x x + 2 (pv + X — be 4- a<£) y x 4- (1 + v 2 + a 2 4- b 2 — X 2 — p 2 — c 2 — cf> 2 ) z x 
+ 2 (ap — bX — c — vcf>) w x , 
kw 2 =2(bv-c/x +a + Xcf)) x x + 2 (cX- av 4- b 4- pcf)) y x +2 (a/x -bv + c + v(f))z x 
+ (1 + X 2 + p 2 + v 2 — a 2 — b 2 — c 2 — cf> 2 ) w x , 
values which satisfy identically x 2 2 + y 2 2 z 2 2 + w 2 2 = xi* + y x 2 + z x 2 + w x 2 . 
Dividing the linear equations by k, and forming with the coefficients on the right- 
hand side of the equation so obtained a determinant, the value of this determinant is 
4- 1 ; the transformation is consequently a proper one. And conversely, what is very 
important, every proper transformation may be exhibited under the preceding form 1 . 
1 The nature of the reasoning by which this is to be established may be seen by considering the analogous 
relation for two variables. Suppose that x^y-y are linear functions of x and y such that x 1 * + y 1 2 =x 2 + y 2 ; then 
if 2£=x + x lt 2tj=y + y lt £, 7) will be linear functions of x, y such that £ 2 + y 2 = £x + yy, or £(£-x) + y(y-y) = 0; 
£-.r must be divisible either by y or else by y-y. On the former supposition, calling the quotient v, we have 
x = £-vy, and thence y = v£ + y, leading to a transformation such as is considered in the text, and which is a proper 
transformation; the latter supposition leads to an improper transformation. The given transformation, assumed 
to be proper, exists and cannot be obtained from the second supposition; it must therefore be obtainable from 
the first supposition, i.e. it is a transformation which may be exhibited under a form such as is considered in 
the text.