104 
Proceedings of the Royal Society of Edinburgh. [Sess. 
+ 01*02* 
cos (p 1 ±P 2 )( u XJ - _ii_U 7 U<, + - 6s i U 8 CT 10 - II U - _ii_U 3 U, 
Sj + Sj l 69 Sj + 2 s 5 ' 9 *,-2*,, 8 10 3 6 Sj + 2s., 3 ‘ 
+ 
2 Sr, 
2sj + s 2 " 4 " 0 ' (2sj - s 2 )(s 
+ cos (Pi-P^ f _ u TJ _ 6s, 
l " 6 " 9 Sj + 2s 
Gspv 
b^io + .-^rtW-U.U, 
S 1 “ 2S 2 
2 s 9 
+ < 2-2 
- - i^T m^> v ‘ n < - sr^ u *°‘ + + ‘- >x “) 
+ - ,, jyw - ^v-®- 
* ^ - »®-®» + A '* - A ®.®. 
- s ^ 5£.-., ) Us1J - +( - +a, - )X “L 
I, iW/ 1 ' 1 ' 7 ' ^V/ : ' r - " X ‘ 8 } 008 + U, ^.i,',, I " '"J iP 
+ terms of the 5th and higher orders in ygq an( l V?- 
(5) 
The terms of higher order in the series may be determined in the 
same way as the terms in <^> 3 and 0 4 , and we thus obtain the complete 
expansion of <j>. 
We may note that instead of assuming (s 1 q 1 — s 2 q 2 ) as the lowest term 
of our integral, we might have assumed q v or q v or any linear function 
of q x and q 2 ; the integral then obtained would be merely a linear combina- 
tion of our integral (5) with the integral of energy, whose lowest terms 
are (s^ + s^)- 
We may further note that in the above process, when finding </> if we 
may if we please add to 0 4 any terms of the form aq ± 2 + /3q 1 q 2 + yq 2 2 , where 
a, /3, y are constants ; for these terms are annulled by the operator 
fs 4 — -f-s 2 — \ and therefore </; 4 satisfies its differential equation just as 
\ dp 1 dp 2 / 
well when these terms are present as when they are absent. The intro- 
duction of these terms into <p i will cause changes in the terms of higher 
order — in <p 5i 0 6 , etc. : and the sum total of all the changes will merely 
amount to adding to our function </> a quadratic function of the two 
integrals which we know, namely, the integral of energy and the integral 
(5) itself. 
Similarly we may add any terms of the form (agq 3 + /3q ± 2 q 2 + y q 1 q 2 2 + dq 2 s ) 
to <p 6 : the ultimate effect is merely to add to our integral a cubic function 
of itself and the integral of energy. There is evident] y nothing to be 
