98 CERTAIN IMPORTANT FUNCTIONS 



corresponding to other than very small values of e q e q f may 

 be regarded as a vanishing quantity. 

 This gives 



> _ A/+V /-ON 



(313) 



or 



^V+^' + ilog(2,)-ilog[-(^)'-(^)']. (3U) 



From this equation, with (289), (300) and (309), we .may 

 determine the value of $ corresponding to any given value of 

 e, when <j> q is known as function of e q . 



Any two systems may be regarded as together forming a 

 third system. If we have F or $ given as function of e for 

 any two systems, we may express by quadratures J^and $ for 

 the system formed by combining the two. If we distinguish 

 by the suffixes ( ) x , ( ) 2 , ( ) 12 the quantities relating to 

 the three systems, we have easily from the definitions of these 

 quantities 



=ff 



(sis) 



$12 | 04 > *f7T7' / p^ 1 fj T7" / n^ 1 ' ^ 2 x7 /O-1 \ 



/ / y 



where the double integral is to be taken within the limits 



Vi = 0, V 2 = 0, and e l + e 2 = e 12 , 



and the variables in the single integrals are connected by the 

 last of these equations, while the limits are given by the first 

 two, which characterize the least possible values of e 1 and e 2 

 respectively. 



It will be observed that these equations are identical in 

 form with those by which F'and $ are derived from V p or cf> p 

 and V q or <f> q , except that they do not admit in the general 

 case those transformations which result from substituting for 

 V p or (f> p the particukr functions which these symbols always 

 represent. 



