32 



GANESH PRASAD, 



Therefore (1) becomes 



Kf\Y'(x, + l, t')-Y'(x„ t')\dt' = cf'^^^lYix', t)-Y(x',0)\dx' + b,,{^E + s), (2) 

 0 a-j 



I I < 1, — 3t < < :7t — 1. 



Therefore, supposing the units to be so chosen that — = 1 , the equation 



c 



/l Y'ix^ + 1, t')-Y'{x^, t')\ dt' ^ /^'"^^l r(x', t)- Y(x', 0)1 dx', (A^) 

 'o 



— Ä < < Ä — 1, 



is an approximation to the fact of the conservation of the energy inside any 0, 



-^{SJE+ s) being neglected. 



Now, from the conservation of the energy inside any 0 of course follows 

 the conservation of the energy inside any closed surface S; but a little reflec- 

 tion shews that the possibility of an approximation to the former does not 

 necessarily involve the possibility of an approximation to the latter. With this 

 tinderstanding (Aj) may be regarded as the first approximate condition of the pheno- 

 vnenon. 



The second actual condition of the phenomenon is this. T^(t) is continuous 

 in t, or, if it has any discontinuities , they are of the second kind ; further , if 

 is a point of discontinuity , T^(tJ is contained in the aggregate of values 

 assumed by T^{t) as t approaches t^. 



Now 



T^{t) = r^,„ t) = r(x,,,„ + e,,A, t), |e,j < i. 



Let e stand for the greatest of the values of the fluctuation of Y(x, t) in 

 the intervals (x^^^ — X, x^^^+l) whatever t may be. Then, if t^, t.^ be any two 

 values of i, 



|G,J<1, |0,J<1. 



Therefore, if T^{t) be replaced by Y{x„ t) in the above statement of the 

 actual condition , an approximation , (B,) , to it is obtained , e being neglected ; 

 here x, Stands for anj' x lying in the interval {x,,^^ — l, x,,^^-{-l). 



The third actual condition of the phenomenon is the impermeability to heat 

 of the faces of the slab. In other words, the third condition is that the quantity 

 of heat which flows across any area 6, situated on any face of the slab, in any 

 interval (0, t) be nil. Therefore , making use of the fact of the conservation of 

 the energy inside any closed surface, it is easily seen that the third condition 

 is equivalent to 



