1200 
MR. R. F. GWYTHBR ON THE DIFFERENTIAL 
Then the nine integrals take the form 
2 /^ — ^ = constant 
y — y^x + ^ a: — = constant 
li^x — — constant. 
]i^ — constant 
= constant 
= constant 
= constant 
+ h.^ = constant 
Ti-^k^ — Sk-^k„k^ — 2k^^ = constant 
(38), 
(39), 
of which the last six do not contain x, y, or y^. 
To select from among these six first integrals those which are functions of the 
matrices only. 
It is readily proved that 
'^3^3 “ ^3j 
Ho ^'1 — 3^05 
f^3^3 — 
■^3^3 — 
2 ^ 3 , 
k^ 
0 , 
and forming the functional solutions of 
we get 
dl^ 
2 /i. 
d]i„ dh^ dk^ 
dh 
die. 
dh. 
3yfco 
- 2A 
h 
kjjc^ “h 
k^k^ ~~ Sk^k^kj^ 
2 L 2 
. (40) 
(41), 
forms which are at once recognized as being obtained from those of the last six inte¬ 
grals (39), by interchange of the subscripts 1 and 3 of h, and of 1 and 4, 2 and 3 of k. 
