String Galvanometer of Eintlioven* 215 



For the complete solution the complementary function 

 containing the arbitrary constants must be added to (25). 

 This takes three forms according to the relative values of 

 the physical constants n s and h„ 



When n 8 >%k» the transient portion takes the oscillatory 

 form 



_h£ 

 <p 8 = A s e 2 cos (njt — ccs) . . . • (29) 

 where 



n.'=(V-W)*, (30) 



A, and « s being the arbitrary constants to be determined 

 according to the initial conditions of the motion. 



When n s < \k Sj the transient portion takes the exponential 

 form, 



<£,=A S / W +B,/ /V , (31) 



where yj and fi" are the two roots of the equation 



p* + k.ii + nf = 0, (32) 



namely 



fl/= -£+*.' (33) 



and 



/«."=- 1*-«,', (34) 



n,' being the same as in (30), but with the signs of the terms 

 under the radical changed, so that nj is a real quantity. 

 When n s = ^k s 



_^ 

 cj> 8 =(A 8 +B s t)e 2 (35) 



As the velocities corresponding to the three cases just 

 given are required for finding the arbitrary constants, they 

 are given below. 



When n 8 >\k„ 



fk 1 -M 



(j) s = —AA s cos (n/t-**,) +W sin (n.'t — a,) > e 2 . (36) 



When n s <-p s 



4.= A.'p,'J*' t +B.p.''J t9 " f (37) 



When n s =\k s 



k s t 



$=[b.-(A. + B^]«~ 2 (38) 



