196 REPORT—1862. 
at once, without explaining how he was led to it, the analytical investigation 
of the substitution in question, viz. in each of the two memoirs he in fact 
writes 
dgsinn+dysin ¢ eee 
dz sin n—dy sin f 
tand¢=f, tandn=g, fy=p, he 
that is 
p=tanlZtand,; g=tan }f+tan }y; 
and in terms of these quantities p, q, the equation becomes 
dp _ dq 
VP VQ 
P=( A+B+4+D)p+2Ep*+(—A—B+D)p’, 
Q=(—A+B—D)q+2E¢+( A—B—D)¢’, 
so that P and Q are cubic functions (not the same functions) of p and q 
respectively ; and the equation for the time is found to be 
where 
dtr 2g _ pdp gdq 
aNa (—pyvP' (1+9)7VQ’ 
which are the formule for the solution of the problem, as obtained in Euler’s 
first and second memoirs. 
47. In his third memoir, viz. that “‘ De Motu Corporis &c.” in the Petersburg 
Memoirs for 1765 (printed 1767), Euler considers the body as moving in 
space, the forces being as before as (dist.)—2._ Assuming that the coordinates 
4, z are in the plane = Ssateammed to the axis, there is in this case 
Zz 
1 
the equation of areas y 77 —* “a =const.; and writing y=y’'sin yy, z=y’ cos yy, 
that’ is, y'= /y?+z2, and y the azimuth, the integral equations for the 
motion in the variable plane (coordinates #, y') are not materially different in 
form from those which belong to the motion in a fixed plane, coordinates «, y 
(see post, No. 56, Jacobi); and the last-mentioned equation, which reduces 
l 
itself to the form y” Ht =const., gives at once dy in a form such as that 
above alluded to (anté, No. 43), and therefore ~ by quadratures. The 
variables employed by Euler in the memoir in question are 
u+u,v—u (say 7, s) and y, 
v, u being, as above, the distances from the two centres, and y the azimuth 
of the axial plane. The functions of v,s under the radical signs are 
of the fourth order; this is so, with these variables, even if the motion 
is in a fixed plane ; but this is no disadvantage, since, as is well known, the 
ease of a quartic radical is not really more complicated than that of a cubic 
radical, the two forms being immediately convertible the one into the other. 
48. Lagrange’s first memoir (Turin Memoirs, 1766-1769) refers to Euler’s 
three memoirs, but the author mentions that it was composed in 1767 with- 
out the knowledge of Euler’s third memoir. The coordinates ultimately 
made use of are v+u, v—u (say 7,s) and y, the same as in Euler’s third 
memoir, and the results consequently present themselves in the like form, 
