4:2* Mr. L. C. Jackson on the Theory of the 



Initial Conditions. 



Let us now consider the form of the general solution for 

 certain initial conditions of starting the oscillations in the 

 coupled systems, and so evaluate the eighteen constants of 

 integration a, b, c, 0, <£, and yjr, each with subscripts *, 2 , 3 , 

 which are not altogether arbitrary, but are such that they 

 satisfy equations (2) for any value of the time. 



Suppose that, at the instant t = 0, the circuits were 

 without charge, but that a current i is suddenly started 

 in the first circuit. 



"We may then write 



e 1 =e 2 =e 3 = 0, h = i , i a =t 3 = for t=0. 



Introducing these conditions into equations (11) and (12),. 

 we obtain 



= a! cos 6 1 + a 2 cos@ 2 + a 3 cos #35 ! 



1 

 = b x cosfa + bzcosfa + b^cosfa, )> . (13) 



= Ci COS ^Tj -f C 2 COS yp 2 + C B cos tys, J 



2*0 = a i( r j ccs #1 + w i ^ #1) + #2(7 cos 2 + co 2 sin 2 ) ' 



-h o^(s cos # 3 + g) 3 sin <9 3 ), 



= 6 X (^ cos <£ x + «i sin fa) 4- ^(r cos fa + «2 sm ^2) 



-f b.^s cos 3 H- ft) 3 sin^3J, 



= ^(^cos^ + aji sin-v/rj + c 2 (r cosi|r 2 -fctf 2 sin i/r 2 ) 



+ c 3 (s cos-v/rg + a) 3 sin ^r 8 ) . • 



(13) and (14) are satisfied by — <^^±^=-, 

 Inserting these values in (14), we have 



/ = a 1 m l -f a 2 <o 2 -f a B a>», 



= />]&)! + 6 2 0O 2 + ^30)3, f" (15) 



= <?!&>! -f r 2 a) 2 -t- ^30)3. ! 



In order to be able to evaluate a, b, and c in terms of the 

 constants of the circuits, we must obtain some other relation 

 between them. The values of o 1? 6 l5 t*i, etc., bear the same 

 relation to one another as the values of A, B, and C in 

 equation (4), and we can easily determine the ratios of 

 A to B and A to C as below, thus providing the necessary 

 supplementary relations required to evaluate a, b, and r. 



It may be noted here that the fact that a x , b u c u etc., are 

 related to each other in the Fame way as A, B, and C 

 gives at once the information that 1 = 4> l — sjr 1 , 2 = (f>2 — y l / '2y 

 ^3 = 03 = ^3? agreeing with the conditions deduced from. 

 (13) and (14). 



i> (14> 



