ON THE SPECIAL PROBLEMS OF DYNAMICS. 213 
the problem of two centres. The case is indicated where b=0, c=0, the 
ratio 6: c remaining finite. 
The case is briefly considered of a particle moving on a given surface. 
105. The third memoir purports to relate to a system of particles, but the 
formule are exhibited under a purely analytical point of view ; so much s0, 
that the coordinates of the points (3 for each point) are considered as forming 
a single system of variables «,, #,,...#;, The partial differential equation is 
de do doe 
—— ) =2(U 
(se) +(e)" +() RU 
which is transformed by introducing therein the new variables p,, p,... p; 
analogous to the elliptic coordinates of the second memoir, The memoir 
really belongs rather to the theory of the Abelian integrals (in regard to which 
it appears te be a very valuable one) than to dynamics. 
Memoirs by Jacobi, Bertrand, and Denkin, relating to various Special 
Problems. 
106. I have inserted this heading for the sake of showing at a single view 
what are the special problems incidentally considered in the under-mentioned 
memoirs which are referred to in several places in the present Report. 
107. Jacobi, “ De Motu puncti singularis ” (1842).—I call to mind that 
the memoir chiefly depends on the theorem of the Ultimate Multiplier (the 
theory in its generality being developed in the later memoir “ Theoria Novi 
Multiplicatoris &c.,” 1844-45). § 4 is entitled “‘ The motion of a point on the 
surface of revolution,” which, the principle of the conservation of areas holding 
good, is reduced to the problem of the motion on the meridian curve, and thus 
depends upon quadratures only. §5 is entitled “On the motion of a point 
about a fixed centre attracted according to a certain law more general than the 
Newtonian one” (ant2, No.85). § 6. ‘On the motion of a point on a given curve 
and in a resisting medium ” (resistance=a-+ be™, or=a-+ bv”); and § 7. “On 
the Ballistic Curve,” viz., the forces are gravity and a resistance=a-+ bv”. 
108. In Jacobi’s memoir “ Theoria Novi Multiplicatoris &e.” (1845), § 25 
is entitled «« On the motion of a point attracted towards a fixed centre” (see 
ante, No. 87); § 26. “On the motion of a point attracted towards two fixed 
centres according to the Newtonian law” (ante, No. 56); § 27. ‘ On the rota- 
tion of a solid body about a fixed point” (post, No.193); § 28. “On the problem 
of three bodies moving in a right line; the Eulerian substitution; theorems 
on homogeneous forces” (ante, No. 91); ‘and § 29, «The principle of the ultimate 
multiplier applied to a free system of material. points moving in a resisting 
medium ; on the motion of a comet in a resisting medium about the sun.” 
109. And in Jacobi’s memoir “ Nova Methodus &c.” (1862), besides § 64 
and § 65, which are applications of the method to general dynamical theorems, 
we have § 66, containing a simultaneous solution of the problem of the motion 
of a point attracted to a fixed centre and of that of the rotation of a solid body 
(post, No. 206), and § 67, relating to the motion of a point attracted to a fixed 
centre according to the Newtonian law. 
110. Bertrand’s “ Mémoire sur les intégrales différentielles de la Mécanique” 
(1852).—¢ III. relates to the motion of a point attracted to a fixed centre by 
a force varying as a function of the distance; §IV. to the case where the 
forces arise from a force-function U= aa “) (or, what is the same thing, 
y . 
