the calculus of functions. 
243 
and by differentiating 
from which may be found. 
This is a solution derived from a certain form attributed to 
<p, but we might also give to <p the form 
<p (x, tya) = \a-\- x p (x—ayf {a, $a, . . $ n a) 
and, in that case, the equation to be solved would be 
yx — + % (x ( x — &Y)f (a, -i/a, Va, . . 4 n a ) ) 
this contains only the second function of the unknown quan- 
tity and must be solved as a second functional equation, con- 
sidering &c $ n a as constant quantities ; let its solution be 
■tyx = F | x, a, yfya, • • } ( a ) 
then we must put x=a and determine -tya from the equation 
tya = F | a, a, -ba, . . ip n a j 
the value of thus deduced, will furnish the values of -if a, 
See. 4 Az, and these being substituted in a } will give the value 
of tyx ; this solution is evidently of a different nature from the 
former, and forms another species. 
Again, the following form of q>. will also agree with the 
conditions 
( p ^ Xytya 'j -^a-y-x^ (*-<!)*/ 4^> 4* •• 4^> 4> •••4 / 
which being substituted in the Problem -tyx must be found 
from a functional equation of the k th order; x must then be 
put equal to a, and the new functional equation of the n rb 
order relative to a must be solved ; this is a third species of 
solution different from either of the former. Respecting 
these three species of solutions, a very important question 
MDCCCXVI. K k 
