34 



GANESH PRASAD, 



^ci^ + Zu^m COS i)tx e , 



a,„ being given by 



2 r'^ 

 J /(«) cos niada. 



7f 



Now tlie conditions (AJ, (BJ, (CJ for t) are practically the same as tlie 

 conditions (A), (B), (C), of tlie previous part, for T{x, t). Tberefore, since f{x) 

 is continuous and f'{x) is finite, it is evident from the results of the previous 

 part that (A,), (BJ, (CJ are satisfied by 



Y{x, 0) = fix), 

 Y{x,^ t) = V{x, t). 



Tliere remains, therefore, only one condition to be satisfied; it is that 

 ^E+s and e be negligible. This condition is a necessary one; for, otberwise, 

 (Aj), (Bj), (CJ would cease to be approximations to the actual conditions of the 

 phenomenon. For the sake of clearness I will call and e negligible 



3 + £ e 



when — - — and — are each less than 10 ^, where g has the same meaning as 

 in Art. 30 and g^ stands for the greatest value of \ Y{x, 0)|. 



Superior Limits of _EJ, e, and e. 



88. Consider E. 



G-oing back to the definitions of E^, E^, and E given in Art. 80, it is easily 

 Seen that, for a circular area, 



_ EM<K\2x,n,+(d+u,)n,\, _ (1) 



E,(d,) = QKX,W+27cK\(d + 2l^)W+SW\. (2) 



Let G be the greater of the two quantities , G^, G^, on the right in the 

 above; then 



x;<G. 



Now, 



Y" (x, t) - -- V" {x, 0 = — S ''^^ cos mx . 



L 



Therefore 



1 



[^m^aje-"''^dt', i. e., 2I«J- 



But 



== Z^J f{cc)cosmKda = — -\(-iyf'(„)- f" (a) cos madal. 



