310 Prof. H. A. Rowland on the Ratio of the Electrostatic 

 between and £ 1? % = a sin bt; v = a b cos bt; 



v ^and2^, x — a! amb(t + t') \ v — a'b cos b(t + t')\ 



„ 2t 1 fmd3t 1 a=a! , einb(t + t"); v = a n b cos b {t + t") ; 



&c. &c. &c. 



At the time 0, £ 1? 2^, &c. we must have : — 



x=0; Vo =a b; 



a sin bt l =a l sin b(t x + 1 ! ) ; v + ft 5 cos bti=a!b cos 6(^ + 1') ; 



a 7 sin 5(2^ + t')=a" sin 5(2^ + 1") ; v + a'& cos b(2t x + /') 



=a"6cos&(2« 1 + *") ; 



&c. &c. 



Whence we have the following series of equations to deter- 

 mine a', a", &c, and t', t n , &c: — 



21,2 = „ 2 . 



a 2 b 2 =v ' 



a ,3 6 2 =a 2 b 2 + V + 2t' a 6 cos 6^'; sin 6 (^ + *') = -j sin 6^ ; 



a" 2 6 2 =a' 2 5 2 + i' 2 + 2v a'b cos 6(2^ + *) ; 



sin 6 (2*! + *") =» ^7 sin b (2^ + *') ; 

 a 



a»%*= a " 2 b 2 + V + 2v a"b cos 5 (3^ + (f) ; 



sin6(3^ + ^) = 47 sm ^(^i + 0; 



&c. &c. 



When ^ is small compared with the time of vibration of the 

 magnet, we have very nearly 



t ! =-it 1} t»=-t u *"'=-§*!, &C. 



a' 2 = 2a 2 (l+cos^ 1 )=W(l-i(^i) 3 ). 

 a" 2 = 9a 2 (l-K^i) 2 )^ 

 a^=16a 2 (l-J(6O 2 ) ? 

 a"" 2 =25a 2 (l-2(^ 1 ) 2 ). 



