e 
: 
431 
This result is useful in questions relating to attraction. 
For if A be an ellipsoid, every point of which attracts an ex- 
ternal point S with a force varying inversely as the fourth 
power of the distance, and if the point S be the vertex of a 
pyramid, one of whose sides is the right line SPP’, and whose 
transverse section, at the distance unity from its vertex, is the 
indefinitely small area w, the portion PP’ of the pyramid will 
attract the point S, in the direction of its length, with a force 
expressed by the quantity 
Loi Qu 
€ - 75) w, Or —5 
and, putting @ for the angle which the right line SP makes 
with the axis of £, the attraction in the direction of & will be 
2w cos 
Bek 
(9) 
Now, supposing the axis of & to be normal to the confocal 
ellipsoid described through S, it will be the primary axis of 
the surface B, which will be ahyperboloid of two sheets ; and, 
the surface being symmetrical round this axis, it is easy to see, 
from the expression for the elementary attraction, that the 
whole attraction of the ellipsoid will be in the direction of E. 
Therefore, when the force is inversely as the fourth power of 
the distance, the attraction of an ellipsoid on an external point 
is normal to the confocal ellipsoid passing through that point. 
Hence we infer, that if u be the sum of the quotients found 
by dividing every element of the volume of an ellipsoid by the 
cube of its distance from an external point, the value of vu will 
remain the same, wherever that point is taken on the surface 
of an ellipsoid confocal with the given one. 
The question of the attraction of an ellipsoid, when the 
law of force is that of the inverse square of the distance, has 
been treated by Poisson, in an elegant but very elaborate 
memoir, presented to the Academy of Sciences in 1833 (Mé- 
