52 



GANESH PRASAD, 



with a given x, T {x, t)' passes from one value to anotlier it assumes all the 

 intermediate values. 



The third coi/ditioii, (C^), is for G t) what (B^) for T{x, t). 



The fonrfii co)idition, 



lim Q t) ^ 0, lim Q (|, 0 = 0, (D,) 



expresses the impermeability of the faces, Q i) being the quantity of heat 

 which flows across unit area, placed at § at right angles to the axis of x, along 

 this axis in the interval (0, 0- 



48. It should be noted that , in the present theory , of the four actual 

 conditions of the phenomenon three, viz., those corresponding to (AJ, (B.^), and 

 (DJ are the same as the actual conditions in the preceding theories. The 

 addition of a fourth condition has been deemed necessary because of the Sub- 

 ordination of T{x,t) to G{i„f) which is thus the dominating factor in the theory; 

 and it should be noted that neither of the conditions (BJ, (CJ uf^ed involve 

 the other. 



The deficition of temperature, as given by (^j) > constitutes the essential 

 feature of the theory. And thus, in order that the problem may have meaning, 

 it is necessary and sufficient that the initial temperature T (x, 0) be not only 

 an even and continuous function of x but also possess a characteristic function 



6'a,o). 



It should be noted that C'(|,^) need not satisfy any conditions other than 

 those necessary to make the conditions (AJ, (BJ, (CJ and (DJ intelligible and 

 these conditions themselves. For example, C"(|,^) need only be finite and inte- 

 grable in t ; thus it may not be possible to obtain an interval of time , ever 

 so small, in which there are not an infinite number of instants at which G'{^,t) 

 is discontinuous in t. Also C" (|, t) may be meaningless for an aggregate of 

 values of f, of zero content ; in particular G' (|, 0) may not exist. 



Characteristic Functions of Fourier's Type. 



49. Let the initial characteristic function G (^, 0) = cp (§) possess an asso- 

 eiate f(x) which is a finite, integrable and even function oi x; 1 will find the 

 necessary and sufficient conditions, that (|) must satisfy, in order that the 

 characteristic function at any siibsequent time be given by % {^,t), where 



being given by 



ß„, = — / 9)(|')cos;i(|'(?|'. 



7C " o 



It should be noted that x{^,t) is defined onli/ for t>0. 



