42 Mr. Ivory on the Method of computing the 
this case ? or, when he says that it is sufficient for the purpose 
he has in view to have reduced the integration of the formula 
■ ‘ to that of the formula are we to under- 
stand that the method of No. 9 is to be extended to every case 
when y is any function of the sines and cosines of the angles 
and m 0 ? 
If the former be Lagrange's meaning, then we must sup- 
pose it to have been his intention to pass over in silence the 
more general case of the question which does not come under 
the method of integration he employs. 
On the other hand, if we are to suppose that Lagrange 
intended his demonstration to apply to the general case when 
v' is any function of the sines and cosines of the angles O' and 
vs ; then it must be owned that this part of his investigation 
is directly at variance with the conclusion drawn in my paper, 
which limits the truth of Laplace’s theorem to the single case 
when 1/ is a rational and integral function of three rectangular 
co-ordinates of a point in the surface of a sphere. But, even 
if this be the sense of Lagrange, it will be allowed that, in 
so nice a case, a proof, which proceeds upon a transformation 
that cannot be performed, is not very decisive : and the fol- 
lowing argument seems to destroy all the evidence of the 
process when it is extended beyond the natural boundary. If 
we integrate v'dq, between the limits <p = 0 and <p = 27 r, and 
put/ vdtp = 2tt . r (o) ; then we shall have, as in No. 3 of my 
paper, 
dy • dp 
~P~ 
27T 
/ n(°). dy 
J p • 
but the method of integration there employed, which is the 
