500 
A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION [153 
or as it may be written, 
(G$£ V, £$B, H, Zy, E, H, Z)j 
+ №& ^ C$s, H, Z)-№ 17, Cjx.y, z)j ; 
or again, 
y-y, ?-*$B, h, z)) Q 
+ №& y> £$H-x, H — y, Z — z) j 
or what is the same thing, 
v -y, r-*$a, H, Z)) 
+ (tr.fl$B-x, H-y, Z-z$fc 77, £)) ’ 
and it is easy to see that the equation will be satisfied by writing 
( il$f 77 -y, S-z)= ( T$f, 77, 0, 
(tr.il$S-x, H-y, Z — z) = — (tr. T]£B, H, Z), 
where T is any arbitrary matrix. In fact we have then 
( 77 —7/, C -*$B, H, Z)= ( T$£, 77 , H, Z), 
(tr.n$B-x, H-y, Z-z$f , 77, ?)=-(tr.T$B, H, Z$f , 77, O 
= -( T$f, 77, ?$s, H, Z), 
and the sum of the two terms consequently vanishes. 
6. The equation 
gives 
?-y, ?-*)=(Tjf, 77, O 
n, f) = (Q$«. 2/, *), 
and we then have 
(O+Tjf, V, ?) = №,. y„ *,)• 
In fact the two equations give 
77, C) = (il$a? + a? / , y + y„ * + *,)» 
or what is the same thing, 
2(|, V, C) = (® + ® < , y + y,, * + *,), 
which is the equation assumed as the definition of (£, 77, £); and conversely, this 
equation, combined with either of the two equations, gives the other of them. 
7. We have consequently 
(»> y, *)=(ii-(0-T)$f, ,, r), 
(?> V, i) = ((n + ^'r , n$jc l , y„ z), 
and thence 
0»> y, *) = (il^(fl-T)(il + T)nil$* # , y y> *,).