272 HENRY A. EOWLAND 



between and t t x =. a sin bt v = a b cos bt 



" t t and 2t t x = a' sin b(t + t'} v = a'b cos b(t + /') 

 " 2^ and 3*, x = a" sin b(t + I") v = a"b cos b(t + t") 



At the times 0, t t , 2t,, etc., we must have 



x = v = a b 



a sin W, = a' sin *(*, + *') v + a b cos W, = a'b cos (/, + t ) 



a' sin &(2f, + t'} = a" sin b(2t, + t") v.a'b cos b(2t, + t") 



etc. = a"b cos *(3f, + J") 



etc. 



Whence we have the following series of equations to determine a', a", 

 etc., and t', t", etc. 



a fi b* = 2 i 2 + v* + 2r a b cos U t \ sin b(t t + t'} = |? sinW, 



" 2 * 2 = a' 2 5 2 4- Vo 2 + 2y a'i cos b(2t, - t') ; sin b(2t t + t") = ^sin i(2/, + /') 



S^ + i!"); sin 4(3^ + /'")= sin J(3/ 4 + r') 

 etc. etc. 



"When t, is small compared with the time of vibration of the magnet, 

 we have very nearly t' \t t \ t" = i fl t'" = f t fl etc. 



a" = 2a \l + cos bt t ) = 4<(1 - t (W,) 2 ) 



fl'" -9a 2 (l-f(^) 2 ) 



a'"* = 16a \l-$(bt t y) 



a iv2 = 25a 2 (l 2 (&,)*) 



T2 = 

 Whence 



a' = 2a (l - 4 (&,)') 



a" =3-/ (l -*(,)') 



a'" =K(1-|(*O*). 



a iT =5fl (l- (d/,)) 



Now a , a', a", a'" and a" are the values of 3 with 1, 2, 3, 4 and 5 

 discharges and a , 2a , 3a , 4a and 5a are the values provided the 

 discharges were simultaneous. 



This correction is quite uncertain as the time, ,, is uncertain. 



In assuming that the impulses were equal we have not taken account 

 of the angle at which the needle stands at the second and subsequent 

 discharges, nor the magnetism induced in the needle under the same 

 circumstances. One would diminish and the other would increase the 



