Theorem in Probabilities. 249 



analogue of (8) is then 



+ ^f^- ) /(^''-^' + 2)+..-+/(^'' + ^'); 



and when ^ is treated as very great the right-hand member 

 becomes 



/(„,,.) {i+^'+^^ '+...+/+ 1} 



+ +/( /i '-2) 2 + l./ i ' 2 } 



The series which multiplies / is (L + l)% or 2^. The 

 second series is equal to /u' . 2^', as may be seen by com- 

 parison of coefficients of x l in the equivalent forms 



(e x + e- x ) n =2 n (l + %a: 2 + . . .) n 



= 6»* + 7^" 2 > + ^"""^ *(—*)* + .... 



I- . w 



The value of the left-hand member becomes simultaneously ■ 



so that we arrive at the same differential equation (10) as 

 before. 



This is the well-known equation for the conduction of heat, 

 and the solution developed by Fourier is at once applicable. 

 The symbol fi corresponds to time and r to a linear co- 

 ordinate. The special condition is that initially — that is when 

 fi is relatively small — -f must vanish for all values of r that are 

 not small. We take therefore 



/(ft*) = -^*-**, (12) 



which may be verified by differentiation. 



The constant A may be determined by the understanding 

 that/(yu, r) dr is to represent the chance of an excess lying 

 between r and r + dr, and that accordingly 



f + 7(ft r)dr=l. ..... (13) 



*/ — °0 



e ~ z ~dz= s/tt, we have 



&~V(h> ■-•■■■ (U > 



