in a series of the attraction of a spheroid. 103 
rent ways, in a series of terms that satisfy the general equa- 
tion in partial fluxions :* and from this it appears, that the 
developement obtained by the algebraic operations described 
above, is identical with the developement found by Laplace's 
method. 
There is another way of expressing the terms of the deve- 
lopement ofy, namely, by definite integrals. But upon this 
head there is no difficulty, when the possibility of the deve- 
lopement is allowed. The question is not concerning the pro- 
perties of a series of quantities that satisfy the general equa- 
tion of partial fluxions ; but whether the developement can 
be admitted in all cases as a fit instrument for the inves- 
tigation of truth. 
Since it has been shown that the developement of y, ob- 
tained by the procedure in the second chapter of the third 
book of the Mecanique Celeste , is the same with the like deve- 
lopement found by immediately transforming the given ex- 
pression into a function of three rectangular co-ordinates ; it 
is just to say of the former method, that in reality it is nothing 
more than a particular way of effecting such a transforma- 
tion, while at the same time it gives to the transformed quan- 
tity a certain arrangement. In the process we have followed, 
there is no need to employ the differential equation that takes 
place at the surface of the spheroid ; and by thus going more 
directly to the foundations of the method, we can discern 
more clearly what goes on under the cover of many compli- 
cated analytical operations. There is an essential distinction 
with regard to the developement to be observed between the 
two cases when the given expression y is explicitly a function 
* Mec. Cel. Tom. II. pp. 32, and 33. 
