444: Mr. Herapath on Functional Equations. 
nounce positively, that the above distinction is universally true. 
Possibly there may exist certain equations, perhaps partially 
indefinite, which may contain in their solutions functions par- 
tially arbitrary, or arbitrary between certain limits ; but I con- 
fess I do not know of any such. 
Though Mr. Babbage and others have solved several equa- 
tions, both simple and porismatic, I do not know that this 
essential line of distinction has occurred to any of them. As 
a proof, however, of the importance of well understanding the 
difference between these two species of equations, I may ob- 
serve, that notwithstanding their solutions are obtained by 
means so very similar, yet the solution of neither species, as 
it appears to me, contains nor is contained in that of the other. 
For example (2),‘which is the solution of (1) when fa, fx are 
indefinite, neither contains nor is contained in the solution of 
the same (1), when this equation is limited to the conditions 
of possibility, namely f2.fav =1and fia = — fa .fiaa. 
By the above considerations the solution of ; 
S {a, ba, Vax, pea, .... }=0, (3) 
taken generally, has but one form, not accounting of course 
for any thing those changes which the double signs of radical 
quantities may occasion. This single solution is in general 
to be deduced from the elimination of Wax, Wa2r,... be- 
tween the equations which result from the successive substitu- 
tion of az, a°2,.. . for x in the primitive equation. 
The same observations extend to 
Sravdpay dPagisntion ba: (4) 
For since }’v can always be equated with ¢a’e— ‘x, in which 
a may have any given form at pleasure, provided only that « 
be confined to the same order of periodicity as y is, we may 
transform, by putting gap a, r) wo 'x,... for Wa, W2,..- 
and then changing z into ¢2, cur (4) into 
SiG 2, Pax, para,.....f=0; 
in which ¢ is the form to be determined. This equation is 
similar to (3), and consequently in a general point of view li- 
mits ¢, apd hence likewise ¥, to one form. It is obviously 
essential in this solution that we have the order of periodicity 
of Y given, otherwise the problem cannot generally be solved ; 
and that « must be of precisely the same order. ‘The gentle- 
man who first treated of the solution of (4) seems not to have 
included the order of periodicity of amongst the necessary 
data; and seems to have imagined that if ax be not taken 
= « its order is not of great consequence, as it would merely 
limit 
