198 REPORT—1862. 
and he then inquires as to the conditions of integrability of these equations, 
for which purpose he assumes that the equations multiplied by mda+ndy 
and pdx +vdy respectively and added, give an integrable equation. 
52. A case satisfying the required conditions is found to be 
B Y=2 PA 
= =, Y=2a4+—= 
x eae i Vf ve 
or, what is the same thing, 
U=2au+2, Vedat 2 ; 
that is, besides the forces a os which vary as (dist.)—2, there are the forces 
2au, 2av, varying directly as the distance, and of the same amount at equal 
distances; or, what is the same thing, there is, besides the forces varying as 
(dist.)—2, a force varying directly as the distance, tending to a third centre 
midway between the other two, a case which is specially considered in the 
memoir; it is found that the functions in r, s under the radicals (instead of 
rising only to the order 4) rise in this case to the order 6. 
53. Among other cases are found the following, viz. :—= 
7. Veen t Sw tie, 
5 
5 
Vau+2 a vw 
2°, Vaan +h ut", 
V=tu+5 v, 
f 
where B=e, or else ce=PSd=2(e. 
In regard to the subject of this second memoir of Lagrange, see post, Mis- 
cellaneous Problems, Liouville’s Memoirs, Nos. 100 to 105. 
54. In the ‘Mécanique Analytique’ (1st ed. 1788, and 2nd ed. t. ii. 1813), 
Lagrange in effect reproduces his solution for the above-mentioned law of 
force (say U= Zt 2yu, Vas +2). There are even in the third edition 
a few trifling errors of work to be corrected. The remarks above referred to, 
as made by Lagrange in his first memoir, are also reproduced (see ante, 
Nos. 49 and 50). 
55. Legendre, “Exercices de Calcul Intégral,” t. ii.(1817), and “Théorie des 
Fonctions Elliptiques,” t. i. (1825), uses p* and q’ in the place of Euler’s p, ¢; 
the forces are assumed to vary as (dist.)—2, and in consequence of the change 
Euler’s cubic radicals are replaced by quartic radicals involving only even 
powers of p and q respectively ; that is, the radicals are in a form adapted for 
the transformation to elliptic integrals; in certain cases, however, it becomes 
necessary to attribute to Legendre’s variables p and q imaginary values. 
The various cases of the motion are elaborately discussed by means of the 
elliptic integrals; in particular Legendre notices certain cases in which the 
* In the ‘ Mécanique Analytique,’ Lagrange’s letters are *, qg for the distances r-+-q=s, 
*—q=w: the change in the present Report was occasioned by the retention of p, q or 
Euler’s variables. 
