in a series of the attraction of a spheroid. 105 
the right hand side, which stands in place of the developement, 
containing all the quantities in y, and besides an infinite num- 
ber of other quantities, denoted by -f- M and — M, which 
have opposite signs, and destroy one another. But this mutual 
destruction of the extraneous quantities will take place only 
when the totality of the series, comprehending its infinite 
number of terms, is taken into account. Any determinate 
number of terms of the developement will contain quantities 
not to be found in jy ; and, for aught that appears, the diffe- 
rence may have any given amount. No finite part of the 
developement can therefore properly be said to represent the 
proposed function. If we take a separate term, as Q w , it is 
not extravagant to say that it may have nothing in common 
with the original expression. Nevertheless such a term is a 
necessary part of the series, in order to balance quantities 
that occur with opposite signs in other terms. 
For the sake of illustration, suppose a spheroid of revolu- 
tion determined by this equation, viz. 
r = a ji +e^i— ^ j ; 
e being a small coefficient, and p the cosine of an arc reckoned 
from one of the poles of revolution. In such a spheroid the 
greatest diameters will make an angle of 45 0 with the equa- 
tor, resembling the figure which the planet Saturn was, some 
years ago, supposed to have. In order to develope the vari- 
able part of the radius of the spheroid, that is in a 
series of quantities that satisfy the equation in partial fluxions, 
it is first necessary to expand V ; and the number of 
terms of the developement will therefore be infinite. 
To the preceding equation let there be added the term 
mdcccxxii. p 
