75]
ON THE ATTRACTION OF AN ELLIPSOID.
435
consider a radius vector on the conical surface such that the cosines of its incli
nations to the axes are
£. §> {e = V(i* + <?+®')l >
P, Q, R and © being functions of the parameter co, and of another variable </>, which
determines the position of the radius vector upon the conical surface. Also let p be
the length of the portion of the radius vector which lies within the ellipsoid; then
representing by dS the spherical angle corresponding to the variations of co and <£, the
attraction in the direction of the axis of x is given by the formula
Also by a known formula
dS= A-AP krr
03
dQ dR dR dQ
i(f) dco d(f> do)
A =
+ Q
P
p ©
dS.
and it is easy to obtain
dR dP dR dP
d<f> dco da> d(p
2 (a© 2
+ R
dPdQ_dP dQ\)
d<f> dco 8co d(f)
IP 2 + mQ 2 + nR 2
The quantities P, Q, R have now to be expressed as functions of co, 0, so that
their values substituted for x, y, z, may satisfy identically the equation of the cone.
This may be done by assuming
P=p,
Q = mb (or + n8) (la + ^ cos </>] — nG s i n </>>
whert
R = nc (co 2 + m8) (la + cos <f^J 4 mb sin 0,
p = (co 2 + m8) (a) 2 + n8) — m~b 2 (co 2 + n8) — w 2 c 2 (ro 2 + m8),
JJ- = m-b 2 (co 2 + n8) + n 2 c 2 (or + m8),
P 2 = (ay + 18) (ro 2 + m8) (co 2 + n8)
l 2 a 2 m 2 b 2
or + 18 co 2 -f m8
n 2 c 2
co 2 -I- n8
a system of values which, in point of fact, depend upon the following geometrical con
siderations : by treating x as a constant in the equation of the cone, that is, in effect
by considering the sections of the cone by planes parallel to that of yz, the equation
of the cone becomes that of an ellipse; transforming first to a set of axes through
the centre and then to a set of conjugate axes, one of which passes through the
point where the plane of the ellipse is intersected by the axis of x, then the equation
P 2 vy 2 . . . 7/ z
takes the form j+ a = 1) and is satisfied by £ = A cos <p, r/ = B sin </>, and - , - being
of course linear functions of these values, the preceding expressions may be obtained.
55—2
