ON THE SPECIAL PROBLEMS OF DYNAMICS. 195 
the same functions of +,s respectively, and m and n being integers (or more 
generally for other relations between the forms of R, S given by the theory 
of elliptic integrals), the equation admits of algebraical integration ; but as 
the relations in question do not in general hold good, the theory of the 
algebraical integration of the equations plays only a secondary part in the 
solution of the problem, It is, however, proper td remark that Kuler, when 
he wrote his first two memoirs “On the Problem of the two Centres” (post, 
Nos. 45 and 46), had already discovered and was acquainted with the theory 
ly nd 
of the algebraic integration of the equation TR Wa (R, 8, m, 2, ut supra), 
although his memoir, “ Integratio zequationis 
da dy 
VA+Bo+Co*+Da%+Ea* VA+By+Cy?+Dy'4 Ey” 
N. Comm. Petrop. t. xii. 1766-1767 ?, bears in fact a somewhat later date. 
45. Having made these preliminary remarks, I come to the history of the 
problem. 
It is I think clear that Euler’s earliest memoir is the one «De Motu Corporis 
ce.” in the Petersburg Memoirs for 1764 (printed 1766). In this memoir 
the forces vary as (dist.)-?, and the body moves in a given plane. The 
equations of motion are taken to be 
oe =2y (—+=), 
u 
ay 2u( Ay By 
” 
—¥ = 97 [| —_ 4 
de vy wp 
which, if ¢, » are the inclinations of the distances v, u to the axis respectively 
~ (See foot-note to No. 42), lead to 
dv? +d? 4gdt? ¢ te B be “="), 
vou a 
v'u* de dn=2gadt (A cos £+B cos n+D), 
where D, E are constants of integration. Substituting for v, wtheir values in 
terms of y, and eliminating dt, Kuler obtains . 
dfsinn P+/P?—Q? 
dy sin Z “4 Q 
3 
where 
A cos n+ B cos +D cos ¢ cos n+Esin ¢ sin 7»=P, 
A cos +B cosy+D =Q. 
And he then enters into a very interesting discussion of the particular case 
=0 or B=0 (viz, the case where one of the attracting masses vanishes, 
which was of course known to be integrable); and after arriving at some 
paradoxical conclusions which he does not completely explain, although he 
remarks that the explanation depends on the circumstance that the integral 
found is a simgular solution of a derivative equation, and as such does not 
satisfy the original equations of motion,—he proceeds to notice that an 
inquiry into the cause of the difficulty led him to a substitution by which 
the variables were separated. 
46. But in the memoir * Probléme, un Corps &e.” in the Berlin Memoirs 
- for 1760 (printed 1767), after obtaining the last-mentioned formule, he gives 
02 
