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) ’