354 
PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
Of these four equations (I,)', (O)', (I3), (IJ, the first will be found to be satisfied 
for the various forms of f tliat satisfy the other three, Ijy the appropriate 
determination of the constant y. to be associated with each such form. Also, (I,,)' 
is the condition to be satisfied in order that (I3) and (Ij.) may possess common 
solutions. To obtain these common solutions, we proceed as follows. 
Let the equations (Ig) and (Ij) be written 
V,./'.:. 0, V,/= 0. 
Then by actual substitution we obtain the results 
Also 
Vyt = 0, 
VjCt 
= 2h 
V,h = a, 
VI) 
= c 
X 
<1 
A 
II 
LX/ 
= 0 
1 
J 
Vyd 0, 
Vwd 
= 21/ 
1 
V,1I = a\ 
VM 
= c! 
i 
( 
V,c' = 21/, 
V.c' 
= 0 
J 
V,V = 0, 
V.W 
= 0. 
Vi/c' = 0, 
V^X' = K, 
vy = 2V - V 7 ’, 
3/r' - V5, 
V./c' = 3X' - Vr, 
V„X' == 2/a' - V5, 
Vo/x' =3 e', 
— 0 , 
Vp- = 0, 
Vp- = — 5, 
and therefore 
Vi,s = — r. 
Vik' 0, 
Vi (V - iVr) = zc', 
Vi (/x' - iVs) = 2 (X' - iVr), 
V/=z 3 (/x'— iV.s), 
We write 
k ' = /•, X' - iV7- = /, 
V^s = 
0; 
Vo/c' 
= 3 (X' - 
- i^r) 
- 
iV7-) 
= 2(g'- 
- W*) 
V,(/- 
iv.) 
= 0. 
/ - \Vs 
= m. 
v' = n\ 
► 
and then these equations give 
= 0, VI- = 3/ ] 
V/ = h, VI = 2m [ 
Vpu = 2/, V.m = n | 
V J = 3z», Vo7^ =0 J 
