90 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
[13 
which are indeed the direct results of the general form above given, the sign (f) 
being placed in succession over the different columns: and the three forms, as just 
remarked, are in this case identical. 
We see from the first of these that u is of the second or third, from the second 
that u is of the first or third, from the third that u is of the first or second of the 
three following forms: 
U = Ho 
1 cl, b, c, 
d 
, u=H, 
a, b, e, f 
, u = H. 2 
a, c, e, g 
> e, f 9, 
h 
c, d, g, h 
b, d, f h 
which is as it should be. 
The following is a singular property of u. 
Let 
, . du ,, . du 
a=tj-, 6= à 
A' = i 
du 
da’ . 2 db ’ 2 dh ’ 
then, u' being the same function of these new coefficients that u is of the former ones, 
u! = u 3 . 
To prove this, write 
p = ah — bg — cf+de, q = (ad — bc), r = eh —fg; 
a,—ap — 2 qe, 
e, = — 2ra + pe, 
b,=bp — 2qf 
f=- 2 rb +pf, 
c,=cp- 2qg, 
g, = - 2rc+pg, 
d, = dp — 2 qh, 
h / = — 2 rd + ph ; 
case of the general formula just obtained 
u/ = (p 2 — 4 qr) 2 
u = u 2 . u, = it 3 . 
a, = h', 
e, = d', 
b, = ~9', 
f,=~c', 
9, = ~V> 
d= e', 
h, = a': 
h" are derived from 
Also 
whence u / = u', that is u = u 3 . 
There is no difficulty in showing also, that if a", b‘ 
a', b' ...h', as these are from a, b, ...h, then 
a" = u 2 a, b" = u 2 b, ... h" = u 2 h. 
The particular case of this theorem, which corresponds to symmetrical values of the 
coefficients, is given by M. Eisenstein, Crelle, vol. xxvn. [1844], as a corollary to his 
researches on the cubic forms of numbers. 
Considering this symmetrical case 
U — aa? + 3 /3x 2 y + 87 xy 2 + 8y 3 , 
u = ql 2 & — Qab/Sy — 3/3 2 7 2 + 4/3 3 S •+ 4^7,