H4f 
Mr. Babbage's essay towards 
may be proposed. What degree of generality does each pos- 
sess, and how many and what sort of arbitrary functions does 
each solution involve ? To discuss this question, and to point 
out the nature of other solutions yet more general, which may 
be found for these and other similar Problems, would far ex- 
ceed the limits of a mere outline of the calculus. I shall con- 
clude my remarks on this Problem by stating the plan to be 
pursued in one particular case, which may serve as a model 
for all similar operations. Take as the form of <p (.r, -tya) 
this is a differential functional equation which must be solved 
on the hypothesis of a , \pa, yf fa, and being constant 
quantities. Let its solution be 
we must now put and treat the resulting equation as 
one of the second order, considering ^ o and ^ o as constants. 
Let its solution be 
Now substitute o for a retaining o as a letter instead of making 
it actually zero, there will result a new functional equation of 
the second order, whose solution is 
and lastly, substituting this value of tyo, and also that of ?o 
which may be deduced from it in (2) we have the value of 
tya, from this may be found, and these being substituted 
in (1) give the value of 
then we have 
