388 Mr. Airy on Laplace’s Investigation of the 
in which case (as the same applies to every other term) the two 
developments are the same. And this is all he offers as a proof 
of this important theorem. 
Now I would ask whether there is for this assertion the 
slightest evidence whatever! and whether all that has been demon- 
strated before, if such vague reasoning may be allowed at all, 
is not entirely opposed to it? It has been clearly shewn that the 
integral of {¥° — Y’°| Z® is always=0 when & differs trom a, 
whatever be the value of Y° — Y’: and why it is so evident 
that it is not = 0 when & =7 it is not very easy to say. If any 
thing but positive demonstration could be permitted, the analogy 
of the preceding cases would tend te shew that this integral 
is = 0 when k =7. For on the ground of generality the function 
Z” may be made even more comprehensive than Z" : if, for stance, 
k = 27, it is well known (by what follows in No. 16.) that the 
number of arbitrary constants in its expression will be about 
twice as great as in the expression for Z, and consequently it 
admits of far more variation of form than Z°. And if in every 
one of these numerous variations of form, the integral of 
\yY° — Y} Z° js 0, much more probable would it seem, that 
as Z admits of far less variation, the integral of {F — Y! Z® is 
constantly = 0. : 
I think it is evident from this statement, that the theorem 
alluded to rests at present entirely on Laplace’s unsupported 
assertion. It is necessary now to consider in what manner it can 
be demonstrated. I have not been able to discover any shorter 
or more general method than the following. 
Suppose y to be a rational function of », /1—«°.cos », and 
/1—.n.sin», of s dimensions. Laplace has shewn (Liv. III. 
No. 16,) that if F be a rational function of 4, ./1—# cos », and 
/1-xn'.sin, it is the sum of a series of quantities represented by 
