220 Dr. A. C. Crehore on the Theory of the 



Making j/ = 0, when t = 0, we find 



tan# s = ; = r . . (53) 



The constants are now determined in terms of the funda- 

 mental physical constants in (51) and (53). y is the 

 deflexion of the centre of the string, and, substituting the 

 constants in (50), we finally have the complete motion 



I^-t) L v 1 t)j 



1 r / ^ 2 \s 



+ - cos I (3V - X e-tan- 



n . 3tt; 



sin 



H 



(54) 



It is evident that when £ = the string takes the form of 

 the arc of a circle, as the equation then reduces to (9). 

 Since the galvanometer is arranged to observe the middle 

 point only of the string, we may put 



. 7r.v ., . 3*7r<r „ 



sm-y=l, sin — j— = — 1, &c. 



the terms alternating in sign. The equation shows that if 



k 

 the tension is large, so that n x is large compared with — , the 



coefficients within the brackets become as 



11 In 



1 : 27 : 125 : 343' ' 



and the second and third terms become appreciable in the 

 record ; but the chief departure from the simple law of 

 logarithmic decrement is due to the second term. The 

 negative sign of this term shows that the string shadow 

 oscillates between logarithmic curves during the latter part 

 of the time which have an initial amplitude greater than 

 those corresponding to the initial deflexion of the string. 



With small quartz fibres, say between *002~and *003 mm. 

 in diameter, the air damping is so great that two or three 

 waves are all that can be seen on the record, when the tension 

 is almost as great as it is advisable to make it. The effect of 



