44 Mr. Ivory on the Method of computing the 
and because x, y, z are constant quantities, there will be the 
same powers and combinations of them in the integral as in 
the fluxion, the coefficients merely being changed: therefore 
and farther it is such a function as, being expanded into a 
series, can coincide only with a rational and integral function 
of the same quantities, consisting of a finite or infinite number 
of terms. Therefore the equation 
cannot take place unless u is a like function of x, y, z. 
The review which we have here taken of Lagrange's 
memoir, and the observations we have made upon it, confirm 
the conclusions drawn in my paper, and throw additional light 
upon this difficult subject. We are indebted to the skill and 
abilities of Laplace for the invention of an equation in partial 
fluxions which has already contributed much to advance our 
knowledge of that branch of physical astronomy which relates 
to the figure of the planets, and which promises still greater 
improvements by suggesting new methods and removing the 
obstacles that have impeded the researches of former mathe- 
maticians : but he has not been so happy in founding his ap- 
plication of this invention on the theorem concerning the 
attractions at the surfaces of spheroids. It is impossible to 
deny that this theorem, as it is delivered in the Mecanique 
Celeste , is unsupported by any demonstrative proof ; and that 
the extent of it has not been well understood. Instead of the 
indirect investigation which Laplace has followed, it were to 
be wished, for the sake of greater clearness and of avoiding 
the subtilties that occur in his analysis, that the attractions of 
the expression \ s -f- a 
ds 
d 7 
is likewise a function of x,y,z; 
