486 
A “ SMITHS PRIZE ” PAPER ; SOLUTIONS. 
[534 
J\? ’ d\' ’ db ’ ^ c ' * S n0t ~ ^ ’ anc ^ three equations are therefore equivalent (as 
they should be) to the equations 
A ^ + (G — B) qr = 0, &c. 
What precedes is a complete answer to the question, but in regard to the actual 
expressions of p, q, r, it may be remarked, that these quantities may be expressed very 
symmetrically in terms of the quantities 
A, p, v = tan 6 cos f tan 6 cos g, tan \6 cos h, 
which determine the positions of the principal axes in regard to the axes fixed in 
space, by means of the angles of position (cos f cos g, cos h) of the resultant axis, and 
the rotation 6 about this axis ; viz. writing k = 1 + + p" + v 2 , we then have 
Kp = 2 ( X' + vp — pv), 
Kq = 2 (— vX' + p + Xv), 
kv = 2 ( pX' — Xp + v), 
and the above result may be verified a posteriori without any difficulty. See Camb. 
Math. Jour., vol. ill. (1843), [6], p. 224, [Coll. Math. Papers, vol. I. p. 33]. 
11. Find in the Hamiltonian form, 
drj _ dH drs _ dH 
dt~ dvr ’ ~df = “ dy ’ &Gm 
the equations for the motion of a particle acted on by a central force. 
Taking as coordinates r the radius vector, v the longitude, y the latitude, the 
equation of the vis viva function is 
hence 
T—\ {r' 2 + r 2 (cos 2 y . v' 2 + y' 2 )}, 
dT 
gf'= r = r suppose, 
dT , , 
= r~ cos 2 y. v = v „ , 
dT 
dy>= r -y = y » ’ 
and the expression of T in terms of r, v, y, and of the new coordinates r, v, y is 
T = %( r 2 + 
whence writing 
+2); 
r- cos y r 
H = h r 2 + 
+ 
r* cos*“ y r 
V,