107 
[122 
aation of a 
nation of a 
term trans- 
msformation, 
>ns. It will 
■ the second 
beorem (viz. 
formation is, 
at that the 
points of a 
the second 
n the same 
лее that an 
эгорег trans- 
i applies to 
that is, to 
ler such that 
I be current 
, Z\, w 1 and 
for shortness 
se 
j coordinates, and 
to a factor pres. 
i when a point is 
; mentioned: but 
proper to explain 
of two variables, 
+ b{£y + yx) + cr]y: 
(a, ...) (x, y, z,ic) 2 
122] A SURFACE OF THE SECOND ORDER INTO ITSELF. 
In fact, these values satisfy identically the equations 
x 2 , 
y*> 
z*, 
w 2 
= 0, 
Xy, 
У и 
z \, 
Wy 
a , 
ß, 
y> 
8 
that is, the point (x 2 , y 2 , z 2 , w 2 ) will be a point in the line joining (x 1} y x , z u Wy) 
and (a, /3, 7, 8). Moreover, 
(a, ...)(x 2 , y 2 , z 2 , w 2 y= (a, ...)(ff 1} y lt z 1} wy) 2 
-fyy (a, • •.)(«> ft % &)(«i, 2/i> ™i) 
+ ^2 ( a > •••)(«. ft r S) 2 
. w 40, 4q-r 
= (a, ...) (x u y lt z u w x ) 2 ~~qi + ~ 2 P> 
that is, 
(a, ...)(x 2 , y 2) z 2 , w 2 ) 2 = (a, ...){x lt y x , z x , Wy) 2 \ 
so that x l} y x , z lf w x being a point on the surface, x 2 , y 2 , z 2 , w 2 will be so too. The 
equation just found may be considered as expressing that the linear equations are a 
transformation of the quadratic form (a,...)(x, y, z, w) 2 into itself. If in the system 
of linear equations the coefficients on the right-hand side were arranged square-wise, 
and the determinant formed by these quantities calculated, it would be found that 
the value of this determinant is —1. The transformation is on this account said to 
be improper. If in a system of linear equations for the transformation of the form 
into itself the determinant (which is necessarily + 1 or else — 1) be + 1, the trans 
formation is in this case said to be proper. 
We have next to investigate the theory of the proper transformations of a quadratic 
form of four indeterminates into itself. This might be done for the absolutely general 
form by means of the theory recently established by M. Hermite, but it will be 
sufficient for the present purpose to consider the system of equations for the trans 
formation of the form x 2 + y 2 + z 2 + w 2 into itself given by me some years since. (Grelle, 
vol. xxxii. [1846] p. 119, [52] (fy 
I proceed to establish (by M. Hermite’s method) the formulae for the particular 
case in question. The thing required is to find x 2 , y 2 , z 2 , w 2 linear functions of 
x i, 2/1, z-y, Wy, such that 
x 2 + y 2 2 + Z 2 -l- W 2 2 = Xy 2 + yy 2 -1- Zy -I- Wy 2 . 
Write 
xy + x 2 =2^, yy + y 2 = 2 V , Zy + z 2 = 2£ wy + w 2 = 2m; 
1 It is a singular instance of the way in which different theories connect themselves together, that the 
formulae in question were generalizations of Euler’s formulae for the rotation of a solid body, and also are 
formulae which reappear in the theory of quaternions; the general formulae cannot be established by any obvious 
generalization of the theory of quaternions. 
14—2