840 
PROFESSOR A. ll FORSYTH OX THE DIFFERENTIAL INVARIANTS 
10. We require expressions for the increments of the derivatives of functions such 
as (f) {x, y), ?/), , . . ; for this purpose, we proceed as l)efore. We have 
+ h, y + k) = f (X + H, Y + K) ; 
and therefore 
coefficient o^ h"k" in expansion of ^ (X + H, Y + K ) 
m ! n ! 
V q P '• 
t s {h‘’h‘ + ( 'ph'-'lfik + dt] 
where U is the coefficient ]f"k" in 
that is, in 
p <12'>-q- 
V V s' V 
V 'i V s 'P''- <l 
wliere in the summation r and s’ do not vanish together and, if either or q be zero, 
the corresponding term ceases to occur. 
Writing 
we have 
df 
V V' 
,■={) s = 0 ^ 
h ^ « + 1 — "h ^ m — r, )i + I — .pt s ) i 
which gives the required increments for derivatives of a function (f). Similarly of 
course for the Increments of the derivatives of all functions similar to <p. 
Note. —Just as in the expressions for the increments of the various derivatives of 
E, h, G, we can replace, in the expressions for the increments of the various 
derivatives of a function (f), the various quantities on the right-hand sides by 
without affecting the values of the first increments. As before, second increments 
are not needed for our purpose. 
II. Ill particular, we liave 
