io6 Mr. Ivory on the expansion 
jTfx sin (p ; q> denoting the variable angle that the circle on 
which the distance from the pole is reckoned, makes with a 
circle given by position ; and the nature of the spheroid will 
be thus expressed, viz. 
r = j i -j- £ ( pV 1 — p sin <p ) j . 
In this case the quantity to be developed must be put under 
this form, viz. 
y. Vi— p +p^=Vi-fS sin <p : 
and the developement will not only consist of an infinite num- 
ber of terms, but these terms will contain an infinite number 
of quantities which arise from the expansion of the radical in 
the denominator, and which are not to be found in the ori- 
ginal function. 
There is therefore a real distinction to be made between 
the two cases when y is an explicit function of three rectan- 
gular co-ordinates, and when it is not. A method of calcula- 
tion which is clear, exact and elegant, when it is confined to 
the first case, becomes clouded with obscurity, if not merely 
symbolical, when it is extended to the other case. To say 
the least, there are certainly great difficulties which are not 
explained ; and if there be any geometers who hesitate, and 
have doubts, they are not without their excuse, and ought 
not to be entirely condemned. 
2. We come next to consider the differential equation that 
takes place at the surface of a spheroid. Of this equation, 
three demonstrations have been published ; one, in the second 
chapter of the third book of the Mecanique Celeste ;* another 
by the same author, not precisely the same with the former, 
* Tom. II. p. 28. 
