338 
PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
not occur if s = 
Writ i no; 
o 
we have 
n + I, and in neither of them may r and s vanish together. 
m ! / u\ n ! 
m 
r 
{in — r)! -z'! ’ j {^n — s )! .s 1 
dt 
r=03=0 
=o\r \s 
m— y-^ 1 , s 
^ ^ W- M c / V,-+l,.sJ-' n-3 
j—0 3=0 \ ' / W / 
V X' M u E' 
5+1 
. o V y PM r, E' 
»'=0 s=0 \ ' / ! 
Proceeding similarly from the expressions for F and G, we find 
cZF 
dt 
^ V /^^M i ^ E' 
" b)-, s + l-*-* J/i —I', •/!—S 
';'=OS = 0 \ 
r / \ s 
+ V V' p'^M /'^M t p 
I “< —< \^,\„ bi-s-^ m — r+l,H—s 
r=0s=Q \' / V’ / 
/ni\ hr 
)'=0 s=0 \ ^ / j 
4- V VM "" W 'M „ Y' 
I \ j \ „ I Vcs-*- '/ii-!"; 'rt-S + 1 
+ S 2 (’")(’;) (f„.,„ +>7,, 
)-=0 s = 0 \ ' , ' ‘ 
and 
, hn\rn\ p, 
+ “* " i )■ / \ o / ^'■+r si'i—i', 'i--' ’ 
.■ = 0s = 0 \ ' / 
— 2 V V 
r=0 s=0 \ 
r / \.s 
^ .E' 
bp-, s+l-*- , 1 -s 
I r'i'd 
~r -J — ^ 
,' = 0 s = 0 
■r / \.9 
^ i', S + I //i — '/•, /i — 3 
'/=os=o 'D’/ 
+ s 
'/=( 
'Hi 
+ :£ E 
”, /»i\ /n' 
1/ 
111 —)■+1, n —s 
Note.—A s we now have the first increment of tlie quantities E„,„, F^n;,, G„i„, and as 
the second increments are not required, the quantities E', F', G' on the right-hand 
sides can he rejdaced by E, F, G, witliout affecting the values of the first increments. 
