648 Prof. J. H. Jeans on the Analysis of 



Law of inverse cube : fir~ z . 



7. We now put n — 3 throughout the foregoing analysis. 

 Equations (7) and (8) at once become integrable, and we have 



= (l+«)-% (22) 



= H^ (1 + " ) ~* tan *' ' * * ' (23) 



in which the value o£ u is now /^/H 2 . If we put a = tan 2 y£, 

 the value of I becomes (cf. equation (10)) 



I = G* sin 2 /5 I cos x cos (% cos /3) cos ( p/JGt~ l cosec /3 tan %)d%, 



and J is similar except that the last two cosines are replaced 

 by sines. 



It is not possible to integrate either I or J or I 2 + J 2 in 

 finite terms ; each integral can be shown to satisfy a differential 

 equation of a known insoluble type. On substituting for I 

 and J in the total radiation we obtain F p as a quintuple 

 integral. One integration (with respect to G) can be effected, 

 but the remaining four integrations cannot be carried out in 

 finite terms. Various attempts to evaluate the integral have 

 persuaded me that it will not agree with experiment for large 

 values of p. 



8. We shall accordingly discuss the form assumed by the 

 integrals I and J when p is large. 



Let us put 



K 



= J 2 cos {ax + b tan %) dx, ... (24) 



Hi 



cos (-ax + b tan %) d K • • • ( 25 ) 



then clearly I and J are the sums of integrals of types K 

 and K'. But from equation (24), K is readily seen to be a 

 solution of the equation 



w= K K> ^ 



and K' is a solution of the same equation with — a replacing a. 

 To examine the case of p large, we need only examine the 



