366 ADAMS— THE PROPAGATION OF [December 2, 



Subtracting the equation for the coil k -\- i from (3) and substi- 

 tuting from (4), we get: 



F..,-(2 + /0F,,+ F.,, = o, (5) 



where 



h = R'K' — L'S'p- ^ ip{L'K' -\- R'S'), (6) 



in which it is assumed that the potentials and currents all vary as 

 gipt^ (5) is a linear difference equation of the second order. By 

 the usual method we put 



and find : 



2 -^ Ji , , 



Let these two values be a and /?, the former with the positive sign 

 of the radical. In general a aad (3 are different, and so we get the 

 two distinct solutions required by an equation of the second order. 

 But for li ^o or h = - — 4 a and ft have the same values. The com- 

 plete solution of (5) is therefore: 



T/^_ = (A^ -j- B^ky^' + (A.^ + B.^k)e'''"-' + {A^ -f- B^k){- 1)/"«' 



-f {A^ + B^k){- I fe^P^ + i:(4 a^ + B^^y^^K 



Since Fo = F„ = o, whatever t, A^^^A., = A.^^^A^^B^^B._ 

 ^=B^=^B^ = o; Ap-{- Bp = o and 



(8) 



(9) 

 (10) 



(11) 



where m is any integer. m=o and m^n are excluded because 

 these give /j = o and h=^ — 4, which are already disposed of. If 

 we take m = n-\- 1, w -j- 2, etc., we get the same series of values 



