30 GANESH PEASAD. 



and 



2 



X,.j+i — X,'j < 2A < — (X,', — X,. J, 



it is easily seen from (1) and (2) that 



HJc = p{l-d^)[f'\Y{x',t + r)-Yix', t)}dx' -2^X,0^£i (t, t + r)] 



^1 



+ 269(^2-^Jß(-^l,«2) 



+ j^oi^.-^'.)^(f,t+r), |e,i, !e,j <i. 



Let cl^ be the greatest value that \ \ can bave whatever be the position 

 of L in the plane containing it. Then, neglecting 



pE^ (Xj, x^, t,t-^T, d^) = 6iJcA, Si(t,t + r) + pc {x^-x,) j (d^ + 2AJ £l(t,t + t) + 3ß (x^, x,) } , 

 the quantity of heat H absorbed hy the cylinder is equal to 



pc/ ' I r(a;', t + %)~ Y(x', t) \ dx', 



whatever be the position of L in the plane containing it. (II) 

 30. Let Ei{dJ, E^{d,^ be the greatest values of 



E^{x^, t,t + t, Jj, {x^, x^, f,t + r, (fj 



respectively , whatever x^, x^, t, t + t may be; further, let E(d) stand for the 

 greater of the two quantities E^^ (d^), E.^ (d^), d being the greater of the two 

 quantities d^, d^. Also let i7j, U^, W, and W stand for the greatest values of 

 Pj (Xj^, t,t + T), F^{x^, t, t+r) , Sl (t, t+r), and ü (x^, x^ , respectively , whatever 

 x^, x^, t, t+t may be. 



Now , it has been tacitly assumed in the last two articles that not only 

 are Y'{x,t) and Y"{x,t) existent and, together with Y{x,t), integrable in t but 

 also 77i, are finite. These conditions are therefore necessary though not 

 sufficient. For, it is evidently necessary that E(d) be not only finite but also 

 negligible; for the sake of clearness, I will call E{d) negligible when E{d)lg is 

 less than, say, 10~^, where g stands for 



K\f'' r(.x', 0)dx'\. 



Tt 



Therefore, in order that a suitable approximation to the quantity of heat 

 which flows across a unit area, or to the quantity of heat absorbed by any 

 cylinder standing on a unit area, be at all possible, it is necessary that the 

 periphery of the area be of such size and shape that 



d < lo-^ 



