420 
Mr. Babbage’s essay towards 
Problem XX. 
Required the general solution of the equation 
F | x, ypcV, 4* ax, 4 ^ fix, . . 4/” voc | —o 
Reduce this by Problem XVIII. to the form of 
F | X , ypX, ^X, . . 4'*# j = 0 
in which shape it may be solved by the preceding problem. 
Problem XXI. 
Given the equation 
F j<r, 4'#»4'« [x,^ fix) J =o 
required its solution. 
Assume a (x, y!/ a) == (p f <p x ( a ) 
hence 4/ a? = «>.-■ (#, a?) (6) 
In the left side of this equation, put for x the left side of (a) 
and in the right side, put instead of x the right side of [a) the 
result is 
4 / « (x, 4^ x) = a l >— 1 (§' f §X, (p'y* <p <r) 
substituting this in the original equation, the result is 
F [.r, (®, <p'/<pir), *■.-■($' /<p®,<p'/* $)*,)} =o 
which by putting <p x for x becomes 
F {fix, a 1 *-* (<p‘#, $fx), fx, $Tx)} = o 
this being an equation of the first order relative to (p may be 
solved as above. 
With respect to the number of arbitrary functions which 
enter into the complete solution of functional equations of 
higher orders than the first, I have little at present to offer; 
the difficulty of the subject, and the wide extent of the enquiries 
