650 Dr. S. R. Milner on the 



a = and x = l. If this be done we obtain 



,- .-__ SI ^ C 1 7TX , 1 3 TTX ) 



(Hr=pf{«» IT + g |0« f7 +....) 



= ^ F !(2«+l)^ C0S ^T • • * (14 ' 



Substituting the corresponding values o£ m and B Hl given 

 by (14) in the summation of (12) we get 



/2» + l 7T\2 



l"-2~T ) D * 2n + lir. 

 (2n + l)~ 



^{l-f-JJi- 



C: 



as the solution for the case in which F(Y) = F and is constant. 

 When F(£) varies with the time, let us suppose first that 

 it is a discontinuous function, composed of a series of 

 suddenly occurring increases or decreases during the intervals 

 between which the function maintains constant values. 

 While F(t) is zero before the instant t = 0, let it suddenly 

 take up the value S F at the time £ = 0, undergo the further 

 sudden increase S X F at the time t = t i , S 2 F at t = t 2 , &c, so 

 that we may write 



¥(t) = X 8 r ~F when t>t r and < t r+1 . 



r=0 



Now the expression 



/2n+l«-\* .. 



"(~2~7j D ^-^ 2w + 1to 



r e cos — 9 — n 



c = Cim F{l-f-M (2^-iP } (16) 



that is (15) with &F written for F, and t — t r for t, is a 

 solution of (7) subject to the conditions (8) and (10) and 

 possessing the property that (instead of (11)) 



- d ° =S r F when #=Q and t>t r . 

 at 



Also the sum of any number of expressions similar to (16) is 

 a solution of the differential equation, consequently the 

 expression 



_ ,2«+} *y D(t _ tr) 2n+l7r? , 



( e cos — 9 T ") 



= ei + i kv\ l-f - £ i - -T^-m — - — - f (17 > 



,-=o ( I t »=o (2n-t-l) J 



