ON THE SPECIAL PROBLEMS OF DYNAMICS. 201 
equations of motion, viz., the problem is reduced to a plane one by means of 
the addition of a force as (ante, No. 56), 
63. Cayley’s “ Note on Lagrange’s Solution &c.” (1857) is merely a repro- 
duction of the investigation in the ‘ Mécanique Analytique ;’ the object was 
partly to correct some slight errors of work, and partly to show what were 
the combinations of the differential equations, which give at once the integrals 
of the problem. 
64, In § II. of Bertrand’s ‘“‘ Mémoire sur quelques unes des formes &c.” 
(1857), the following question is considered, viz., assuming that the dynamical 
equations 
ie dU Wy dU 
| df de’ dO dy’ 
have an integral of the form 
a=Px?+Qe'y'+Ry?+8y'+Te' +K 
(where « is the arbitrary constant, and P, Q...K are functions of w and 
y), it is required to find the form of the force-function U. It is found that 
U must satisfy a certain partial differential equation of the second order, the 
general solution of which is not known; but taking U to be a function of 
the distance from any fixed point (or rather the sum of any number of such 
functions), it is shown that the only case in which the differential equations 
for the motion of a point attracted to a fixed centre of forces have an inte- 
gral of the form in question is the above-mentioned one of two centres, each 
attracting according to the inverse square of the distance, and a third centre 
midway between them, attracting as the distance. 
The Spherical Pendulum, Article Nos. 65 to 73. 
. 65. The problem is obviously the same as that of a heavy particle on the 
surface of a sphere. 
I have not ascertained whether the problem was considered by Euler. 
Lagrange refers to a solution by Clairaut, Mém. de l’Acad. 1735. 
The question was considered by Lagrange, Méc. Anal. Ist edit. p. 283. 
The angles which determine the position are y the inclination of the string 
to the horizon, ¢ the inclination of the vertical plane through the string to a 
fixed vertical plane, or say the azimuth. And then forming the equations of 
motion, two integrals are at once obtained; these are the integrals of Vis 
Viva, and an integral of areas. And these give equations of the form 
— dt=funct. (p) dp, dg=funct. (W)dy ; so that ¢, » are each of them given by a 
quadrature in terms of , which is the point to which the solution is carried. 
It is noticed that may have a constant value, which is the case of the 
conical pendulum. 
66. In the second edition, t. xi. p. 197 (1815), the solution is reproduced ; 
only, what is obviously more convenient, the angles are taken to be 
y, the inclination to the vertical, 
@, the azimuth. 
It is remarked that will always lie between a greatest value a and a least 
value f, and the integrals are transformed by introducing therein instead of 
y the angle o, which is such that 
cos L=cos « sin?a+ cos f cosa, 
