246 Mr KELLAND, ON THE MOTION OF 



10. Now, by reference to (7), it will be seen that the conditions of 

 equilibrium will be satisfied by making M = — PQ, or supposing, in 

 analogy with Coulomb's hypothesis, that if the mutual action of similar 

 particles be of one nature, that of opposite ones will be of another. 



The two equations in a, a then become 



5| = PoA.^a - Q^B.Aa, 



~ = - PaA./^a + Q^B. A' a, 

 at 



writing A and B for the functions of x. 



Now if both series of particles vibrate, we must have 

 a = a cos {ut — kx), 

 a = a' COS (u't — kx + c); 



.-. Sa = — 2a. sin^ — - (omitting terms which vanish\ 



, . ,kSx 

 Sa =^ — 2 u . sm^ -— - ; 

 2 



Aa = a' cos {u't - /e(x + Sx) + c} -a cos (ut — kx) 



= rt'{cos (u't - kx + c) cosA'^.a; + sin (tit — kx + c) sin ^^.r} 



— a cos (ut - kx) = a cos . klx — a, 



Aa' = a COS (ut — kx - kSx) — a COS (u't - kx + c) 

 - a {cos (ut — kx) cosA'^j' + sin (ut - kx) sin ^^:r} 

 - a cos (u't — kx + c) = a cos kSx — a ; 

 hence by substitution : 



^ = -2PaA .sin=— .a - Q^B. cos kSx. a' +aQ:^B 

 at 2 



= -2Pcr^.sin^^.a - Q2^.a'+2Q2^.sin^^.c.'+aQ2£ 



= - 2P<.A . sin^ ^.a + 2Q2^ . sin^^.a', 

 since 2.B = 0. (see page 244.) 



