PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
Inserting the values of the invariants that are already known, we have 
B . 2B — = ^ B-‘ 
(In 
and therefore 
I (rfo, w"o) 
y4 
V* p 
</B _ B > 
dn o"!' 
Secondly, let y, g, h _ L, M, N, so that u becomes zch. Proceeding in the same way 
we find ’ 
(ys') = + V-J {ivp, Wo) - J j 
Inserting the values of those invariants whicli have already been obtained, we have 
(after a little reduction) 
J ^ g3 d /' M _ 2B2 dB 
yi 
dn \p' 
T ds 
Hardly, let/, g, h _ a, />, c, so that u becomes w"o. Proceeding in the same wav 
as for Wo, we find 
(I ( V//^ \ 
dn ( V^') "" '" 2^1 - |^cyV + IJ (,e„ u^g). 
Now we have retained I {wo, tvdo) in our aggregate, in place of H (?e"o), so that the 
latter must lie removed from the foregoing expression : as the relation ' 
, J~(tCo, w'\^ = Wow''.^! [iVo, 2v"o) — iryH (le'h) — a/'.PV~ 
holds, we have 
'dn \Y^) ~ «’"o) — w’"yV~ — P (wo, w"o). 
Inserting the values of those invariants which have already been obtained, and 
reducing the equation, we ultimately have 
nyA) = _4B3y(B) + 8Bqff+yBav. 
it may lie noted, in passing, that the above equation, which gives the relation 
letween I (wq, w"o) and H (w'h), leads to the expression for H {zv"o) in the form 
H {w"o) _ B t/B 
yo ~ _// ,7A ~ 
dn 
38. Again, it is known that 
dB 
Kd,J' 
y, SH ^ (JP - 2FQ + EU,^ aK ^ pp I 
jf = GQ - 2FB + ES, = nQ - 2ME + LS 
I . 
