z6 
Mr. Ivory on the Method of computing 
and if we first substitute for -j the series equal to it ; next 
multiply by a .y' . d\jJ . drs * ; and then integrate ; we shall finally 
obtain 
r ci 
20 
■ff 
5 u (2) + Sec . } 
(r — a) a 1 . y\ dyJ. d>m' 
| r 2 =-2ra . y+a* | * 
now by the formula (E) the value of the integral on the left- 
hand side, when r — af\s = 27 ry : therefore 
W = u (0) 4. 3 u (,) + 5 u (2) + 7U <3) + &c. ; 
a formula which is equivalent to what Laplace has deduced 
in his manner,* and which is the foundation of his very inge- 
nious method. In effect, if we develope the given function y, 
as Laplace has taught us to do,-f into a series of terms every 
one of which shall satisfy his equation in partial fluxions ; so 
that 
y — Y (o) + Y (,) + Y (2) + & c.; 
then, since it is proved that this expansion is unique, by 
equating the like terms of the two values of y, we shall have 
generally, 
4 t.Y (!) = (a; + i).U (0 ; 
by means of which all the quantities U , Sec. which 
are the coefficients in the series for V, will become known. 
This analysis proves in the clearest manner that Laplace's 
method is exact only in one hypothesis fory, and that it is 
strictly confined to one class of spheroids : for it can hardly 
be maintained that the formula (E) will be true whatever 
function, the symbol d may be supposed to denote. 
* Liv. 3e, No. ii. f Liv. je. No. 16. 
