Prof. Ayrion and Mr. Mather on Galvanometers. 377 



value consistent with the desired accuracy, e.g., the 1 in the 

 first column indicates that the instrument is only just sensi- 

 tive enough to indicate the maximum permissible error in 

 balance ; the 2 in the first column denotes that the instrument 

 is twice as sensitive as absolutely necessary ; and the 3 that it 

 has three times the necessary sensibility &c. The numbers in 

 line 1 below the various values of m a show the relative times 

 required for the spot to move perceptibly, the time occupied by 

 the slow-moving instrument in moving over the minimum 

 perceptible distance being taken as unity. In column 2 line 1 

 we find the number 1*11, and this indicates that if the short- 

 period instrument is only just sensitive enough to show the 

 maximum permissible want of balance, then, doubling the 

 sensitiveness by making the period \/2 times as great, would 

 enable the want of balance to be detected in a shorter time, 

 the times being in the ratio of 1 to 1*41. Similarly for 

 column 3 line 1, increasing the period to twice that of the 

 short-period instrument and therefore increasing the sensi- 

 tiveness four times would reduce the relative times to the 

 ratio of 1 to 1*5. When m equals 100 (or the sensitiveness 

 is increased ten thousand times, if such an increase were 

 possible) the relative times become 1 to 1*63, showing a com- 

 paratively small extra diminution in time required to detect 

 a want of balance, although the sensibility is very largely 

 increased. 



The numbers in line 2 show that when the short-period 

 instrument is twice as sensitive as is absolutely necessary the 

 gain in time required to detect the want of balance becomes 

 less, the ratio of the times being 1 to 1'06 when m 2 equals 2, 

 and only reducing to a value 1 to I'll when ?ri 2 equals 

 10,000. Subsequent lines show that when the short-period 

 galvanometer is several times more sensitive than is absolutely 

 necessary to obtain the desired accuracy, the gain in quickness 

 in detecting a small error by weakening the control becomes 

 unimportant, and so the small gain will be more than neu- 

 tralized by the longer time taken in the return to zero. 



Aperiodic Motion. 



If the motion of the suspended system be aperiodic, as in well- 

 damped d'Arsonvals, the problem is rather more complicated. 



When damping forces act on a vibrating system the 

 equation of motion is 



dW 2Bd6 G a n 



aW + TdF + T d =°> 



