34 Mr. Ivory on the Theory of the Figure of the Planets 
Again, for the sake of brevity in writing, let us put 
1 1 
@ a Ff Oe 5 
then we shall have 
1 d : , 
Y= [e+ 24 2). aty'dd sind da’. (4) 
For this latter formula is no more than the first one written 
differently, as will be manifest by performing the differentiation 
of g with respect to a. 
It is to be observed here, that Lagrange conceived that the 
formula (4) contained the whole of Laplace’s demonstration, 
without its being necessary to add any limitation whatever re- 
lating to the thickness of the molecules near the attracted 
point. For he says explicitly * that the function 
de 
e+2a7 "> 
is always identically equal to zero on account of the evanescent 
factor it contains ; whence it would follow that the integral 
Se + 2a <<) ay'd¥ sin da! 
must be equal to nothing, whatever y/ stand for, and not equal 
to 4a7ay as in the formula (3), and as he himself actually 
found to betrue. ‘There is therefore an inconsistency between 
the reasoning of Lagrange and the result of calculation ; and 
it is this which he calls wne difficulté singuliére, and a paradox 
in the integral calculus. Now all this arises from not observ- 
ing that the function mentioned is not in every case equal to 
zero. It is so, indeed, for every point of the surface of the 
sphere except one, when cos) = 1; in which case the function 
has an evanescent divisor which balances the evanescent factor 
and produces a finite value. If one element of an integral have | 
a finite value, the integral itself must be a finite quantity; and 
this is the plain and short solution of the difficulty. If La- 
grange’s attention had been directed to the formula 
Se + 2a <) a*(y'—y) di sini! da’; 
and if he had observed that Laplace limited his theorem to the 
case when y/—y is divisible by the evanescent factor which ap- 
pears in the denominator when the molecule is very near the 
point of contact of the two surfaces, there would have been 
neither difficulty nor paradox. But although he would in this 
manner have avoided inconsistency, he would not have ob- 
tained the most general demonstration of the theorem. For 
* Journ, de ? Ecole Polyt, tom. viii, p. 62. 
SS 
AV rt —2r a cos p+ a? 
this 
