108 Proceedings of the Royal Society of Edinburgh. [Sess. 
q 2 = l 2 - hy|- U 3 cos_p 2 + - 1 U 4 cos (2 P 1 + P 2 ) ~ 1 11 5 COS (2pj_ -p 2 )j 
v <5.2 + S.> j 
“ ^{sTTlff 7 C ° S ^ “ T-Tg^ys cos (%2 -JPi)} 
- U 9 cos p 2 + -i- U 10 cos 3p_ 2 1 
+ terms of the 4th arid higher order in y / / 1 and ^/h 2 . 
We know from the general theory of Dynamics* that the expressions 
thus found for q x and q 2 must be the partial differential coefficients with 
respect to p ± and y> 2 of some function of (l v l 2 , p v y> 2 ) : and, in fact, we have 
obviously 
aw 
where 
0W 
( h ~ o j — 5 
dpi dp 2 
W = l 1 p 1 + l 2 p 2 - Ipf J-U! sinp 1 + i-u 2 sin 3 p^ 
- Z,y(J.U 8 sin p., + 1 — U~ 4 sill (2 Pl + p 2 ) + pi — U 5 sin (2p, -p 2 ) ) 
~ l ih{— U 6 sin Pi + 0 . 1 . U, sin (2 p. 2 + Pl ) + 1 — U 8 sin (2 p, -p t ) } 
l ^2 aSc) ~r & — j & .) S 2 J 
- Z 2 f ( — U 9 sin p 2 + — U 10 sin 3p 2 j 
V. 8.^ 05.) J 
+ terms of the 4th and higher orders in Jl L and Jl 2 . . (9) 
The terms in which p 1 and p 2 occur otherwise than in the arguments 
of trigonometric functions are 
p^l i + terms of the 4th and higher order in x // 1 and Jl 2 ) 
4* ^2(^2 4" 55 55 55 55 ) 
Denote the coefficients of p 1 and p 2 in this expression by a x and a 2 re- 
spectively : express l x and l 2 in terms of a x and a 2 by reversion of series, 
and replace l x and l 2 throughout in the series (9) by these values in terms 
of a x and a 2 ; so that we now have 
W = a 1 p 1 + a 2 p 2 - ap (~ Dj sin jo 1 -c U 2 sill 3/?^ 
- + 2^4— U 4 sin (2 p L + p 2 ) + 2 ^ 1 _ So U o sin ( 2 Pi - Pa)} 
- ",V ! 1 U 6 sin^j + 1 U 7 sin (2p„ + Pl ) + — 1 U 8 sin (2p„ - jj,)} 
“ a 2 -{yU 9 sin p 2 + ^-U 10 sin 3p 2 j 
•f terms of the 4th and higher orders in Ja x and Ja 2 . . . (10) 
* Of., e.g., Analytical Dynamics, § 121. 
