CONSTITUTION OF MATTER AND ANALTTICAL THEORIES OF HEAT. 



39 



may be regarded as tlie sufficient and necessary condition. In reality it is 

 only a quasi-necessary condition but less restrictive than (PJ. To see this 

 point let US go back to Arts. 28 and 29. Now, making use of tbe result (1) 

 of Art. 34, it is easily seen that (I) holds even when the quantity neglected 

 is smaller than pE^{x^, t, t + r, dj. In fact E^(x^, f, t+T,d^) may be replaced 

 by, say, 



kK j f P, (x„ t,t + t) + {d^ + 3iA,) P, {x,, t,t + r)\. 



Also, it is evident that, in Art. 29, E^{Xj, x^, t, t+r,d^) may be replaced by 



2 



a smaller quantity in which, for example, c{x2~xj — £l^,(t,t+T) takes the 



place of c(x2 — Xi)x2X^£i(t,t + r),Slg being less than £1. 



It may be mentioned here that, in the light of these remarks, the sense 

 in which the word „necessary" is used in the beginning of Art. 30 and in 

 Art. 32 is easily understood. 



The really sufficient and necessary condition can be obtained in terms of 

 and f(x) provided that definite suppositions be made relating to the ex- 

 pression of X^, s, and A in terms of Aj ; for example, it may be supposed that 



A, = 10- i„ 



^ = -1., 



s * 



and 



A = 3-^xlO-^xA^ 



But, in the face of the fact that the sufficient and necessary condition 

 rests on suppositions which are of a purely tentative character and any change 

 in which may at once degrade it into a quasi-necessary condition, it is evident 

 that, for the purposes of this essay, a quasi-necessary condition is quite suffi- 

 cient. In fact , in a professedly inexact theory like the present one , the im- 

 portant thing is the nature of the restriction imposed on f{x). And, as (PJ 

 clearly shows, this is purely arithmetical. 



Illustrative Examples. 



39. The following simple examples suffice to illustrate the salient features 

 of the theory: 



(i) Suppose that A, = 10-^A,, y < A, < -^^^ and A = 3"^ X 10"^ x A^ 

 Let Y{x, 0) = f{x) = x\ Then the left side of (P,) is equal to 



-^x43|rWI = -^A,<13iA, 



27t/ « 



Now, if JET stand for the total quantity of heat in any right circular cy- 

 linder with its bases of unit area, on the faces of the slab, it foilows from 



