28 GANESH PRASAD, 



where Z" indicates summation with respect to a single row of the assemblages 

 An,,,, ^„„, K is a constant called the thermal conductivity of the slab, and k 

 Stands for 1 or Icj^ according as 3\ lies or does not lie in the interval 



Then it is easily seen that 



where <?i is the sum of the areas of the bases of the rows. Therefore the 

 quantity of heat given by (1) is equal to 



t-\-t t-{-t 

 p{l- c\) IcK \ -f Y' (x„ t') dt' + f \Y' (x„ t') I dt' \ 



t 't 



- (1 + 26, A J (l + e, • -|] p (1 - J A ^ { / "l Y' (x^, t') I dt'+ %,X,P,{x^,t,t + t)\ 

 where p is the greatest value that (?, can have when L is of a Standard shape, 



6 . 



say circular, and d^ stands for 1 and evidently depends on A, the position 



P 



of L in the plane containing it , and the size and shape of the periphery of L. 



Let d^ be the greatest value that ] d^ \ can have, whatever be the position 

 of L in the plane containing it. Now let P^(x^,t,t + t) stand for 



j \Y'{x^,t')\dt'- 



2 . — 1 



further, suppose that > y and Aj<, say, 10"^ Then, if (l^<^, — a con- 



dition which will be shown to hold when L is a circle, — it is easily seen 

 that, neglecting 



pE, {x, ,t,t + t, H,) = plK\ 2Aj P, {x,, t,t + x) + (Ij + P, {x^, t,t + x)\, 

 the quantity of heat ivhich flows across the unit area is equal to 



-phKf Y'{x„t')dt', (I) 

 't 



whatever be its position in the plane containing it. 



29. I proceed now to find the quantity of heat which is absorbed by a 

 cylinder, with its axis parallel to the axis of x and its faces, of unit area, 

 x = x^, X = x^, in any interval (t, t+t). 



Let H represent this quantity ; also let L stand for one of the faces. 



Now, there may exist assemblages, near the surface of the cylinder, each 

 of which is sometimes partly or wholly inside the cylinder, none remaining 

 wholly inside it throughout the interval; it is easily seen that the portion of 



