336 
NON-LINEAR EQUATIONS OF THE FIRST 
(22). 
From the form of the above equations it appears that if 
a and b are so determined as to satisfy two of the conditions, 
f= 
df «¡fy df cfy 
^ dx dx ’ dy dy 
(23), 
they will satisfy the third. For suppose they satisfy the first 
two, then the system (21), (22) may be expressed in the form 
V’ %■ %> D = °> F f V’ * % > if) = 0 - (24)> 
in which the truth of the third equation of (23) is involved. 
Now, as (19) satisfies (18) whatever constant values we 
assign to a and b, it still will do so if, after the differentiations 
by which and ~ are found, we substitute for a and b 
J dx ay 
any functions of x and y. 
But a and b can be determined so as to satisfy two con 
ditions. Hence they can be determined so as to satisfy the 
system (23). Differentiating the equation f— % on the hypo 
thesis that a and b are functions so determined, we have 
d f df da d f db _ d% 
dx da dx db dx dx ’ 
d f df da df db d% 
dy da dy db dy dy ’ 
Here. have the same values as in (23), being ob- 
dx dy ° 
tained by differentiating as if a and b were constant. Hence, 
reducipg by (23), we have 
d f da df db 
da dx db dx 
d f da df db 
da dy db dy 
(25).