the Attractions of Spheroids of every Description. 23 
We have here followed very closely all the steps of the 
demonstration contained in the Mcsanique Celeste , and on first 
thoughts no reasoning can be more convincing, or appear more 
free from all obscurities. This much at least is certain, that 
every part of the demonstration is- placed beyond the reach of 
all objections except the valuing of that term in the equation 
(B), which is derived from the difference between the sphe- 
roid and the sphere : and about this a deeper consideration of 
the nature of the functions concerned may raise in the mind 
some doubts and scruples. No better way can be devised for 
trying the soundness of Laplace’s procedure, than to perform 
that part of the calculation which is alone liable to suspicion, 
without omitting any of the terms which he has tacitly re- 
jected ; to throw out such only as on examination can be 
proved to be necessarily evanescent when < 5 r = 0 ; and to re- 
tain the rest if there be any of a different description. Now, 
to apply this rule, we have/ 2 = 2p 2 (1 — y) ; and j n = (p -J- <5>j* 
— 2j0 ( p -j- ^ r ) y + p 2 — { 1 + • 2 p 2 ( 1 — y) + ^' 2 ; therefore 
f 2 — S/ ' = { 1 + 7} - f- consequently, j = j x { 1 "F 7 } " 
and, by multiplying by dm and affixing the sign of integra- 
r •) Z 
* 1 1 J 1 j *> ani *’ ky expanding the second radical into a 
1 I 
f' f 
series, the complete value of will be equal to 
J f f 
p dm pdm 
tion, the complete value 
will be equal to 
