Prof. Ayrton and Mr. Mather on Galvanometers. 379 



Similarly 



,?r t 2 t* ~| 



6'—a'— — aV 2 - — 2/? j + terms involving £ 4 &c. jo and q' 



Hence the ratio of the two displacements is 



l~ t 2 t* ~l 



O'-ol' an!' 1 L|2 ~ 2y) | 3 +terms in * 4 &c -7 >and /J 



— a a ^ 2 r £ 2 ? I ' V 



Li o "" %>■ 77 + terms in £ 4 &c. p and </ 



but a' = ni 2 a, 



and n '2 = J: 2 



t 4 



Thus, when , - and higher powers are small, the ratio is 



unity, consequently there is no difference in the rapidity of 

 detecting want of balance when the sensitiveness of the short- 

 period instrument is sufficient for the purposes of the experi- 

 ment. 



To find the ratio in the other cases v e may observe that 

 the general term in t r is, when r is odd, 



-f«r-l) W ,, - 1)(r I 2)(r " 3) J> ? + ■ ■ ■ + ('-!>«->. 



and when r is even 



+£{0- 1)^- 2 + (? '" 1)( ^ 2)l =Vv+ . . . +?-*}. 



Decreasing the control decreases ^ and therefore increases q, 



q 2 =zp 2 —n 2 . 

 Hence q is > q, , 

 consequently the term in t* in the numerate - of -r will be 



greater than the corresponding term in ihe denominator, and 

 this indicates that the displacement of the slow-moving 

 instrument gains on that of the quick-moving one as the 

 time increases. 



The above investigation teaches us that when we have a 

 galvanometer whose control can be readily altered and whose 

 sensitiveness can be easily made ample for the purpose of 

 the test, then for rapid working adjust the control so that the 

 sensitiveness is, say, two or three times as great as is absolutely 

 necessary for the desired accuracy* 



