240 REPORT—1862. 
addition) three arbitrary constants; these are ¢,, Y,, p. Writing in the first 
place V=W-+tt,+yy,, the resulting equation for W may be satisfied by 
taking W, a function of ¢ and 0, without y or ¢; and it is sufficient to have 
a solution inyolying only a single arbitrary constant. This leads to a solu- 
tion which is as follows,— 
= 6 
V=tt een tan cee ad aah 
it Wy, y, oe V 12-02 
% ) $,0 
+p) tan Sa tants | 
| ev p—y,7—8,7 eve =o aes 
6+) J J Ow an (Gy 140" e424) 
where ¢, and 6, are certain given functions of ¢,, J, p, and of @ and ¢. The 
solution of the dynamical problem is then obtained by putting the differential 
ts » TB? a dV equal to arbitrary constants L, a, G respectively ; 
the eagle are be ee more simple than might be expected from the very 
complicated form of the function V. The foreg: going results (although not by 
themselves very intelligible) will give an idea of the form in which the ana- 
lytical solution in the first instance presents itself. 
203. Richelot proceeds to remark that the solution in question, and the 
resulting integral equations of the problem, may be simplified in a peculiar 
manner by the method which he calls “‘ the integration by the spherical tri- 
angle.” For this purpose he introduces a spherical triangle, the sides and 
angles whereof are 
coefficients —- 
v,r\, ph; N, A, M, 
and then assuming 
N constant, M=x—9@ 
((-3) sin? (9+») sin’A+ (<-3) cos” (@+v) sin’A = eth , 
where p and ¢, are constant, the solution is 
V=t,t—p(p—A) cos N—pM+p [cos Ad(¢+yr) ; 
and that this expression leads to all the results almost without calculation. 
204. I have quoted the foregoing results from the Note (Crelle, t. xlii.), 
having seen, but without having studied, the Memoir itself: the results appear 
very interesting and valuable ones; but they seem to require a more com- 
plete geometrical development than they have received in the Memoir ; and I 
am not able to bring them into connexion with the other researches on 
the subject. 
205. The solution, §3 of Donkin’s memoir “On a Class of Differential 
Equations &c.” (part i. 1854), is given as an illustration of the general 
theory to which the memoir relates; it contains, however, some interesting 
geometrical developments in regard to the case (A=B) of two equal moments 
of inertia. I have not compared the results with those in my Note of 1849. ; 
206. The solution of the rotation problem, § 66 of Jacobi’s memoir “ Nova 
Methodus &c.” (1862), has for its object to show the complete analogy 
which exists between this problem and the problem of a body attracted to a 
