PROF. A. C. IMXON OX RTTTRM-UOUV1LLK MAI.'Mo.MC KXPANsloNs. 



42H 



Also \ l2 il\//ila, except where a is near its inaxiniuin, and in that part <>f (lie path 



tin- factor exp( O./UL) is so small that the contribution to the integral is negligible. 



** i 



I lence F (/, if, It) is at most // I exp (-a/u) da, that is, -. 



" ft. 



This is the first of Hoi WON 's conditions. 



Ag.-tin 



In the first term of this expression the first term of the subject of integration 

 contains the factor 



which is of the order of expa(x a), that is. at most exp ( a/*) when we take 

 ./ <-/*. Also X~ 1/2 d\ // da as before and X""*//^' 1 *. Hence the contribution of 



this term is // y= at most, and the same is true of the second part of the first term. 





In the other term the integral of jo-j is finite, and the integration with respect to t 

 is over a finite range, so that these two elements do not affect the order of magnitude : 

 the factor ^ (x, 0) ^ (t, 1 ) -=- x* ( 1 , 0) is of the order of X' 3 * expa(z-<), that is, IT 3 '* 

 at most, even when // = : the length of path in the X-plane is 2xlt. Hence the 

 contribution of this term is //It" 12 independently of /u if a-Sa. 



On account of the symmetry tatween .r and t, like results can Ixj deduced if 

 .<> b + n. 



Thus the second and third of HOBSOX'S conditions are fulfilled. 



17. Again, so long as u-^a the value of \fs(x, 0)*(a, l) -i- *(l, 0) is limited, and 

 so is that of \^(x, 0)*(fc, l) -r- *(l, 0). Hence the first term on the right in (9) is 



limited, the integral of 



d\ 



being 2ir. The second term has been found to tend to 



zero when It is increased, and therefore 



F(,*,R)c& 



.'a 



is limited if x^a < If, or similarly if ,rZzl>> a. When >i < jc < // we may write 



*-J>f, 



so that j F(f, .r, R) dt is the sum of two terms, each limited, and is itself limited for 



a 



all values of a, b, x, R. This covers one of HOBSON'S further conditions ( 4 of his 

 paper, p. 361). 



