y 



String Galvanometer of EintJioven. 211 



Putting z 2 =— x 2 + las, and expanding the radical into in- 

 finite series, we find 



-i 2 _ ** 1.3£ 6 

 y "2p 2Ap z 2.4. 6> 5 



Since p is always a very large quantity compared with z, the 



maximum value of which is -^when x = ^ , all terms after the 



first in the series are of the second order of smallness and 

 may be omitted, giving 



=!(**-* 2 ) (8) 



This is the equation of the parabola which most closely 

 approximates the arc of the circle throughout the length of 

 the string. 



The equation (8) may be developed by Fourier's series * 

 into the following, which will be used as the equation of the 

 string when deflected by a steady direct current to determine 

 later the constants in the equations of motion, 



^=-,yo|_sin- r + 3 i 8in- r + 5 -,«n — + .... J, • (9) 

 where — p is written instead of x in the previous equation (8) . 



The Differential Equations. 



We will next form the differential equations applying to 

 the string under the conditions of the string galvanometer. 

 Let y denote the deflexion of an element of the string at the 

 distance x from the end and time t, and Y the impressed 

 force. The partial differential equation f of motion of the 

 string is then 



g + ij.rfJJ+Y (10) 



at* at d,i- K 



*^The developments of x and or by Fourier's series from which (9) is 

 directly derived are 



x = 2( sin x — ^ sin 2x + ~ sin 3x - -. sin 4r + . . . . J 



a 2r/7T 2 4\ . 7T 2 . , /7T 2 4\ . 7T 2 . , 



* = A\l ~ V) Smr ~ 2 9m 2v H 3 ~ sO Sm 3 *~ T Sm4a: 



+(f-i) 



■J' 



t Rayleigh, « Theory of Sound,' 1894 Edition, page 192. 



P2 



