524 PROFESSOR KELLAND ON THE VIBRATIONS OF 



by integration. 



sin a sin 2 a sin 3 a a 3 



and p 22 ga &c. = a log a-a + Ca ^ + &c. 



which vanishes when a=0. 



cos a cos 2 a cos 3 a a? a? a 2 _ a 2 ^ a> 



-p-+ 23+ 33- + &c. = -g toga j- ^- + C- 2 - + D-2gg + &c 



r>. sin a sin 2 a sin 3 a a 3 , a s 



Finally, _ 14 - + ^_ + ___ + & c . = _log + Aa 3 + Da- ^ + &c. 



where A is a constant. 



1 sin a n a 2 , a 4 



d /I sin a n\ a a , a 



J5 (a^-^-)-6 + 3 loga + 2A - 



l a 



d a \a da^a * 



By substitution, we obtain, 



2 _47Tm fl a 2 /I jS^ \ i 



np"if~w" V3~i8o ) } 



,, . 4 7T OT A 2 



Hence in value, c 2 = - 



and 



180 



4 7TWZ 



180 e 45 e 



This is precisely of the/o?-m which experiment requires. We learn from the 

 result, too, that m is negative, that is, that the force is repulsive. 



It will be remarked that whilst approximation, on the hypothesis that the 

 principal effect is due to the particles in the immediate vicinity of that under 

 consideration, gives the same order to the first term in c 2 as the process of summa- 

 tion does, it gives it a different sign. 



This wiE appear by expanding either equation (1) or equation (a) in terms 

 of A. 



From the former <?=1 2 m -- 



A 2 

 = 15 r 



