Statistical Relations of Radiant Energy. 653 



certainly limited function of y, and the capability of mani- 

 pulating it to secure the values given for T and V ! A 

 series of the above type is uniformly convergent in the 

 whole range of values considered if the <£'s steadily decrease 

 to zero, but this in itself is not sufficient to ensure the 

 validity of the expressions for T and V. These expressions 

 are only valid if they are convergent, a condition which is 

 however secured if 



u n being the nth term of a convergent series 



u l + u 2 + ... +u n + .... 



These conditions are also sufficient to secure the uniform 

 convergence of the Fourier series for y and y, and are 

 therefore generally applicable. They are, however, much 

 too general and indefinite to help us to a definite formula, 

 and we must therefore attempt to obtain more precision. 

 The procedure, an arbitrary one, is to apply the general tests 

 of convergency one after the other until a satisfactory solu- 

 tion is found ; the simplest suitable test is probably the 

 correct one. The first test of convergence is obtained by 

 comparison with the geometric progression, and it shows 

 that the above conditions are satisfactorily verified if we 

 can determine a positive number ja(< 1) such that 



02 n ^A(l-/a)» nV^ A ( 1 -/*)". 



where A is some constant number. The range of possible 

 values of: fi being 0< fx *c 1, we see that the possible range 

 of values for </>,?, n 2 (j) n 9 , at least for all finite values of n, 

 however large, is defined by 



< $,? < A, 



< n 2 <\>,? < A. 



In other words, there is no restriction except that the 

 quantities must remain finite. There is therefore in such a 

 case nothing to prevent the application of the theorem of 

 the equi-partition of energy as has been done by Bayleigh 

 and Jeans, and the result should hold good for all attainable 

 regions of the spectrum. This test is therefore unsuitable. 



The second test of convergence is obtained by comparison 

 with the first of the logarithmic series, and it shows that the 



