CONSTITUTION OF MATTER AND ANAMTICAL THEORIES OF HEAT. 



57 



55. The following is a simple method of approximation which is applicable 

 to an extensive class of continuous functions T^(x): 



Let q>Ji) be tbe continuous function of |, to which T,{x) is the limital 

 function. Then two cases arise: the initial characteristic function 6'(|, 0) = (p^(i,) 

 is either admissible or inadmissible. 



i. When C (|, 0) is admissible — and this is the case , for example , if 

 T'f(x) exists and is finite — , an approximation may be furnished by T(x, 0); and 

 it is easily seen that 



\T(x,0)~T,{x)\<£l, 0^x<7t, 



Sl being the greatest of the values of the fluctuation of T^(x) in intervals, of 

 length 21, taken anywhere in the interval (— 7t, n). Let g^ stand lor the greatest 



value of 1 T^ {x) j . Then the method may be said to fail when — is not ne- 

 gligible. 



ii. Let iT, {x) stand for P„ {x) or P„ (— x) according as x is positive or nega- 

 tive, respectively ; also let qCjC^) continuous function of | to which TI^{x) 

 is the limital function. Then, when C(|, 0) = 93i(^) is inadmissible, the function 

 T{x,()), defined by the admissible characteristic function 0) = ^^i^) , may 

 be taken as an approximation to jT. {x) ; and it is easily seen that 



\T{x, Q)-T,{x)\<d + ^', 



Sl' being the greatest of the values of the fluctuation of n^(x) in intervals, of 

 length 21, taken anywhere in the interval (— tt). Since Sl— 2d ^zSl' <: Sl + 28, 



Sl 



the method may be said to fail when ~ is not negligible. 



Illustrative Examples. 



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

 of the theory: 



(i). Suppose that the discriminating aggregate G of the slab is (sr, B^), 



V 



standing for the aggregate of all the rational numbers ^^^^ , with odd deno- 

 minator, in the interval (0, te). 

 Let 



C(|,0) = 9>(l) = 2 5>2, |>0; 



then (p{l) possesses an associate t\x) = fii^)) fii^) standing for the f(x) in (ii) 

 of Art. 24. 



At any point |, D (|, a;') «xj 1 and, consequently , \V' {x, t)^^^^OK>l. Also 



Abhandle, d. K. Ges. d. Wies, zu Göttingen. Math.-Phys. Kl. N. P. Band 2,4. 8 



