202 REPORT—1862. 
by which substitution they assume a more elegant form, involving only the 
radical 
V 1+ (cos /3— cos a) cos 2, 
where & is a constant depending on cos a, cos 3; and the integration is effected 
approximately in the case where cos }—cos @ is small. 
M. Bravais has noticed, however, that by reason of some errors in the 
working out, Lagrange has arrived at an incorrect value for the angle &, 
which is the apsidal angle, or difference of the azimuths for the inclinations 
a and: see the 3rd edition (1855), Note VII., where M. Brayais resumes the 
calculation, and he arrives at the value @=7(1+ 3a), a and 3 being small. 
Lagrange considers also the case where the motion takes place in a resist- 
ing medium, the resistance varying as velocity squared. 
67. A similar solution to Lagrange’s, not carried quite so far, is given in 
Poisson’s ‘ Mécanique,’ t. i. pp. 385 et seq. (2nd ed. 1833). 
A short paper by Puiseux, ‘‘ Note sur le Mouvement d’un point matériel 
Tv 
5 
68, The ulterior development of the solution consists in the effectuation of 
the integrations by the elliptic and Jacobian functions. It is proper to re- 
mark that the dynamical problem the solution whereof by such functions 
was first fairly worked out, is the more difficult one of the rotation of a 
solid body, as solved by Jacobi (1839), in completion of Rueb’s solution (1834), 
post, Nos. 186 and 197. 
69. In relation to the present problem we have Gudermann’s memoir “ De 
pendulis sphericis dc.” (1849), who, however, does not arrive at the actual 
expressions of the coordinates in terms of the time; and the perusal of the 
memoir is rendered difficult by the author’s peculiar notations for the elliptic 
functions*. rk 
70. It would appear that a solution involving the Jacobian functions was 
obtained by Durége, in a memoir completed in 1849, but which is still un- 
published ; see § XX. of his ‘Theorie der elliptischen Functionen’ (1861), 
where the memoir is in part reproduced. It is referred to by Richelot in 
the Note presently mentioned. 
71. We have next Tissot’s ‘Thése de Mécanique,’ 1852, where the ex- 
pressions for the variables in terms of the time are completely obtained by 
means of the Jacobian functions H, ©, and which appears to be the earliest 
published one containing a complete solution and discussion of the problem. 
72. Richelot, in the Note ‘“‘ Bemerkungen zur Theorie des Raumpendels ” 
(1853), gives also, but without demonstration, the final expressions for the 
coordinates in terms of the time. 
Donkin’s memoir “On a Class of Differential Equations &e.’”’ (1855) con- 
tains (No. 59) an application to the case of the spherical pendulum. 
73. The first part of the memoir by Dumas, “ Ueber die Bewegung deg 
Raumpendels,” &. (1855), comprises a very elegant solution of the problem of 
the spherical pendulum based upon Jacobi’s theorem of the Principal Func- 
tion (1837), and which is completely developed by the elliptic and Jacobian 
functions. The latter part of the memoir relates to the effect of the rotation 
of the Earth ; and we thus arrive at the next division of the general subject. 
sur une sphére ” (1842), shows merely that the angle ® is > 
* The mere use of sn., cn., dn. as an abbreviation of the somewhat cumbrous sinam., 
cosam., Aam. of the ‘Fundamenta Noya’ is decidedly convenient. 
