the calculus of functions. 
421 
to which it would lead, induce me to postpone it until I have 
more time for the consideration. The following remarks may 
suffice for the present to point out some of its difficulties and 
the mode of enquiry. 
4'# =/ {#, a, b, See. | 
1 
(0 
^x==f ja?, a, b, &c.~j 
(*) 
2 f f 
4/ 3 x ==/ |<r, a, b, &c. 
(3) 
Sc c. See. 
4/»a?=/{a?, a, 6 , Sec. } 
(») 
n 
From this by eliminating n — \ of the arbitrary constants 
a, h, See. we have an equation of the form 
F | a?, -tyx, oc , . . 4” ® } = 0 (a) 
and it might possibly be concluded that equation ( 1 ) con- 
taining n— 1 arbitrary constants is the general solution of this 
last equation : but this is by no means the case. In the first 
place between the two equations ( 1 ) and ( 2 ), more than one 
arbitrary constant may be eliminated, thus let 
from which we find 
\}/ 8 x = x 
the two quantities a and b have been eliminated, and it is pos- 
sible to select a value of 4/a?, between which and -if ac an infi- 
nite number of arbitrary constants could be eliminated. 
But waving this objection let us consider the case of ( a ) 
which is deduced from the elimination of n — r i arbitrary func- 
tions. 
We have seen in Problem VI. that a function of the first 
