Particles illustrating the Line and Band Spectrum. 447 
mutually attracting particles have already engaged the atten- 
tion of Maxwell in his discussion of Saturn’s rings: the 
equations here to be considered are nearly of the same form 
as those given by Maxwell, but they can be conveniently 
deduced by means of Lagran ge’s equation. 
Let the radius of the undisturbed orbit be @ and the 
position of any two particles (1) and (2) subtending an angle 
20 at the centre be given by polar coordinates P33. "2, 
such that 
P< (1 = Pils ee + po), 
$,=s +o@t+o,, bce Re eae 
2) =ah, fy = ah. 
The radial and angular disturbances are given by p and o 
respectively, and the transversal displacement by ¢. 
The potential of particle (2) at point 7, $,, 2, 1s given by 
€ é 
Vp=— ae 
9 Jr + 2 — Irv, cos (by SVE (ayes 2) 
pee | 
‘e sin? @ © 4 sin? 6 





Since 

Pio = 2a sin? [1 pi ps + Biba + . 
we find by expanding ¢ 
Pi 1 a fe Pir Pe + P1P2 {P1— Ps he eS 
ro 2asin 6 +3(e,+p.)?— pes BA 
f 3 = 2 
‘ f,_=09 cot 0+ pea (1 +2 cot? 0) e (3) 
By simple differentiation, 





Beis 2 6, ee 
Or, adp, 4arsin?d L a 
eee 0 (p, + ps) a 
_ OVie SAE AL: eae ecosO { ae 3p; + p2 ae (A) 
Od, | a(1+pi;)do0, 4a’ sin?@ | 2 | - -(4) 
+ = °? (tan 6+ 2 cot @) } 
ve ee Nig) abs — 61) . 5 
0-; oF Sa? sin’ 9 E 
neglecting small quantities of second order. 
