ON THE SPECIAL PROBLEMS OF DYNAMICS, 205 
(falling bodies, the pendulum, and the gyroscope) are, in reference to the 
proofs they afford of the rotation of the Earth, considered as well in an experi- 
mental as in a mathematical point of view. The second part of the volume 
contains the theory (after Laplace and Gauss) of falling bodies, that of the 
pendulum (after Hansen), and that of the gyroscope (after Yyon Villarceau) ; 
and the whole appears to be a complete and satisfactory résumé of the experi- 
mental and mathematical theories to which it relates. 
85. We have also Cohen “ On the Differential Coefficients and Determinants 
of Lines &c.” (1862), where the equations for relative motion are obtained in 
a very elegant manner. The fundamental notion of the memoir may be con- 
sidered to be the dealing directly with lines, velocities, &c., which are variable 
in direction as well as in magnitude, instead of referring them, as in the ordi- 
nary analytical method, to axes fixed in space. The memoir is a highly in- 
teresting and valuable one, and the results are brought out with great facility ; 
but I cannot but think that the great care required to apply the method cor- 
rectly is an objection to it, if used otherwise than by way of interpretation of 
previously obtained results, and that the ordinary method is preferable. 
I may remark that the theory of relative motion connects itself with the 
lunar and planetary theories as regards the reference of the plane of the orbit 
to the variable ecliptic, and as regards the variations of the position of the 
orbit; but this is a subject which I have abstained from entering upon. 
Miscellaneous Problems. Article Nos. 86 to 111 (several subheadings), 
Motion of a single particle, 
86, Jacobi, in the memoir “ De Motu puncti singularis” (1842), notices 
(§ 5) the case of a body acted on by a central force which is any homogeneous 
function of the degree —2 of the coordinates ; or representing these by 7 cos ¢, 
y sin g, then the force is caky where ® is any function of the angle ¢. In 
i 
fact, after integrating by a process different from the ordinary one the case of a 
central force a he remarks that the method in fact applies to the more 
general law of force just mentioned. 
87. Jacobi, in the memoir “ Theoria Novi Multiplicatoris &c.” (1845), con- 
siders (§ 25) the case of a body acted on by a central force P a function of the 
distance, and besides by forces X and Y, which are homogeneous functions of 
the degree —3 of the coordinates (#, y); viz. the equations of motion are in 
this case 
Ma Px 
de Tp FM 
Cpaoury 
ge hy 
and there is an integral 
rat "aly 2 farteY —y? ps . 
3(ay'’—a'y) —f (wY—yX) a const 
(the function under the integral sign is obviously a function of the degree 0 
in (a, y), that is, it is a function of Y) if X, Y are the derived functions of 
xe 
a force-function U of the degree —2 in (w, y), then there is, besides, the in- 
tegral of Vis Viva, and thence a third integral is obtained by means of the 
