350 Mr. 0. Heaviside on the 



expresses the effect due to the steady impressed force Y at z 1 

 at time t after it was started. This will have a term corre- 

 sponding to a zero p (due to the infinite increase of Si in the 

 previous problem), expressing the final state. Hence, leaving 

 out this term, the summation (134), with sign changed, and 

 £=0, expresses the final state itself. Thus, taking V =l, 



S Au = S^i^/pM 



is the expansion required to be applied to (133). Put 

 A=Wi/pM. in it, and it becomes 



V=2(w/M)eP*f , f'«7i«e-P'i(fo 1 rf* 1 , . . . (135) 

 Jo Jt 



fully expressing the effect at z, t, due to the impressed force e, 

 a^ function of z x and t ly starting at time t . To obtain the 

 current, change u to w outside the double integral. The M, 

 when the condition VC = at the ends is imposed, is that of 

 (130) ; the u and w expressions those of (126), But if we 

 regard S, R', and 1/ as constants (or functions of z), then 

 (135) holds good when terminal conditions V = ZC are im- 

 posed, provided the impressed force be in the line only, as 

 supposed in (135). 



When the impressed force is steady, and is confined to the 

 place 2=0, and is of integral amount e 0) (135) gives 



Y = e Q 3,uwJpM—e 2,uw f*'/pM, . . . (136) 



w being the value of w at 2 = 0, as the effect at time t after 

 starting e . The first summation expresses the state finally 

 arrived at. 



Again, in (135) let the impressed force be a simple har- 

 monic function of the time. I have already given the solution 

 in this case, so far as the formula for C is concerned, in the 

 case Y = at both ends, in equation (76), Part II., which may 

 be derived from (135), by using in it iv instead of u at its 

 commencement, putting e = e sin nt, and effecting some reduc- 

 tions. The Y formula may be got in a similar manner to that 

 used in getting (76), but it is instructive to derive it from 

 (135), as showing the inner meaning of that formula. Let 

 in it e = e sin (nt-\-u), where e is a function of z. Effect the 

 t x integration, with t = Q for simplicity. The result is 



v=- S - 



pt ( p sin u + n cos a\ C l 7 

 — ( A t-, — 2 I 1 uheAz x 



3 v p 2 +n 2 ;j ^° 



^ u /p sin (nt + a)+n cos (nt + a)\C l 



S+ Mr y+tf )J o *w^ as?) 



