146 Mr. Boor on a certain Multiple Definite Integral. 
eur Bu? aft Cw 
SES +A i ee ee NYS ny ee ye GRY ED 
In the other we ma me a et 
er case we may assu (ana u. 
It remains to notice a remarkable class of integrals, the values of which can 
be assigned in finite algebraic terms. Mr. Cayley was the first to point out their 
existence, to the discovery of which he was led by performing the operation 
a2 a a2 
ae Ga eae 
on the function which expresses the value of the integral 
\f Chak ahs 
[Gq =4,)) + G2)? - + F (Gata) ]P 
and observing that the single integration which it involves became possible. We 
may remark, that to this class belong the integrals which express the attractions 
of homogeneous ellipsoids, when the law of force is any even inverse power of 
the distance, except the law of gravitation. Shall we say that this is an exception 
to the observed rule, that the constitution of the material universe, and the pos- 
sibilities of analysis, are arranged in a certain mutual harmony, or shall we regard 
it as a fact subservient to some higher principle ? 
We shall only consider the case of an ellipsoid’s attraction when the force 
varies as the inverse fourth power of the distance. None of the results have, we 
believe, been given before, and as they are of a very interesting character, we 
shall dwell upon them at some length; i. e. we shall consider the attractions of 
ellipsoids of uniform and of variable densities, on both external and internal 
points. 
Now the expression for the attraction parallel to the axis 2, in this case, is 
dedydz(a—2) f (E+ 4 er a eae er 6 
iS: [(a—a) + (by) ae 2 es tas 
dardydz f (+4 +4) 
[faa + (by) (e—2)" 
where 
