538 
ON A PAIR OF DIFFERENTIAL 
[483 
^ is given by M. Delaunay only as far as m 5 , the additional terms of - and expression 
for p were kindly communicated to me by Prof. Adamsj ; and 
v — t 
+ sin 2D (li m 2 + ff m 3 + m 4 + m 5 + to 6 
4- 102859909 m 7 i 7596606727 m 8 _ 8051418161 w 9\ 
' 1244160 ' ^46 4^0 0 1,0 ~lTTW4400~/ 
+ sin 4 -° (!M + fit w 5 + + tMMF +-friWirWt 0 -- 3 - ™ 8 ) 
+ sin 6D (ffif m 6 + ftfflt m 7 ) 
(Delaunay, t. n. pp. 815, 836, 845). 
To integrate the original equations write 
p = 1 + pi + p. 2 + ..., 
V = t +V 1 +V 2 + ..., 
where the suffixes indicate the degrees in the coefficients k, j conjointly: the equations 
for p n , v n take the form 
— _ q _ 9 dv n , y __ r\ 
dtdt pn Z dt +Vn ~^ n> 
d (dv n n i jj 
it \dt +2pn+u ’ 
=p, 
where V n> U n , P n , Q n do not contain p n or v n . From the second equation we have 
d ^ + 2 Pn + U n = il u +jp n dt, 
where fì n is a constant of integration, the integral jp n dt containing only periodic 
terms ; and then adding twice this to the first equation we have 
It Tt + Pn + V <‘ + 21u * = 2n ** + Qn + 2 fp. 
dt 
which determines p n ; and substituting its value in the other equation we have 
and thence v n ; the constant Zl n is determined so that 
dv n 
dt 
dv n 
di' 
may contain no constant 
term. We have 
II 
© 
o' 
il 
fc? 
U 2 — 2 p 1 
f ==- 2 is- 2 4 ! ^ 
(dv 1 y 
[dt) 
U 3 — 2 p 1 
- 2p " W + 6p,Ps - 
&c. 
&c. 
dvx 
di 
dv 2 
dt 
dv x 
dt