NOTES ON THE ABELIAN INTEGRALS, &C. 
367 
58] 
each of which contains only a single root of the equation fx = 0. But in Richelot’s 
second memoir “ Einige neue Integralgleichungen des Jacobischen Systems Differential 
gleichungen” (Grelle, t. xxv. [1843] p. 97), the equations are obtained by direct inte 
gration in a form not involving any of the roots of this equation ; the method employed 
in obtaining them being in a great measure founded upon the memoir just quoted 
of Jacobi’s. The following is the process of integration. 
Denoting the variables by x x , x 2 ... x ny and writing 
so that 
Fa = (a — Xj) (a — x 2 ) ... (a — x n ), 
F Xi — (x x Xy) .. . ( X x X n ), 
&c. 
then the system of differential equations is satisfied by assuming that x l , x.,... x n are 
functions of a new variable t, determined by the equations 
dxi _ V(/ar 1 ) „ 
dt F'x x ’ 
dx 1 
(In fact these equations give S f . — dt S nr = 0, &c.) 
/ v\jX) xx 
From these we deduce, by differentiation, 
d*xi = i d fx x V(.Ai) v/ v'(.A) 
dt 2 2 dxi (F'xx) 2 F'x x ~ {xj — x) F'x 
(where S' refers to all the roots except x x ) and a set of analogous equations for 
j ^3 • • • • 
Dividing this by a—x x , where a is arbitrary, and reducing by 
(a — x 1 )(x 1 —x) 2(a — x)(a — x 1 ) 
1 - 
x + x x — 2a 
X-, — X 
we have 
that is 
1 d 2 x x _ 1 d fxj 
a — x x dt 2 2 (a — x x ) dx T (F'xj) 2 
, ! V(A) V' A,A) _ VP) V(,Ai) (x\ + x-2a) 
2 (a — Xi) F'x x (a — x) F'x 2 “ F'x F'x x (a — x)(a— x x ) (x x — x) ’ 
1 _ 1 f/ 
a — x x dt 2 2 {a — x x ) dx x {F'x x f 
I 1 VQi) V V(.A) _ x .A _ 
2 (a — ¿Ti) A#! ^ (a — &) F'x 2 (a — #j) 2 (F'x x ) 2 
_ IS' V(A) V(.Ai) A + « - 2a) 
2 F'x F'x x {a — x) (a — a^) — ¿c) ’