346 Dr. C. Chr'ee on Longitudinal 



and to a first approximation 



J\ (aX) 2p 2 X 2p 2 X 2 _2 m(m-n) 



Ji(ajj,) a\fjb 2 —p 2 ) <z 2 fi(fjb 2 — p 2 ) fi n{3m—n) 9 



m—n J (aX) X 2 J/(aX) _ 2(m— n) X 2 _ 2m(m — n) 



2nfjb Ji(a/ji) a 2 /n J / (aft) 2n/j, afi fin^dm — nf 



and similarly if a be replaced by b. 



We thus see that the third column in the determinant (38) 

 is such that each principal term in it is obtained by multi- 

 plying the principal term in the same row in the first column by 

 the same constant 2m (m ~ n)-h- \/jbn(3m — n) \. Now if one 

 column of a determinant is obtainable by multiplying another 

 column by a constant that determinant vanishes. It is thus 

 at once clear that in proceeding even to a second approxima- 

 tion we need retain only the principal terms in the second 

 and fourth columns of (38). This removes what seemed at 

 first sight a formidable obstacle, viz. the occurrence of the 

 logs in the expressions for Y , Y x , and Y/. 



§ 13. For further simplification of the determinant multiply 

 each term by 2, and divide the first and second rows by a/n 

 and bfi respectively. Then for the second row write the 

 difference between the first and second rows, and for the 

 fourth row the difference between the third and fourth rows, 

 and multiply the resulting rows by b 2 /(a 2 — b 2 ). Finally 

 multiply the second column by a 2 fi 2 /2, the third column by 

 [Aa?, and the fourth column by a 2 fji<z 2 /2. We thus reduce (38) 

 to the easily manageable form 



i-ia,y -l _^>! 2 (i_i^) -M, 



[M Z —p~ K ° ' fl 2 —p 2 



xyb 2 2 P 2 



1 A2..2 1 



8 T H^-p 2 ) fi 2 -p 2 



1-faV 1 x» + ^^-^(^ + W , ?=5) 1 



%hY 1 ib 2 X 2 (3X 2 + 2a 2 ^) 1 



After algebraic reduction, use being made in the secondary 

 terms of the first approximation results (1), (40), &c, we 

 easily deduce from (41) 



-l 



(4] 



whence 



»-**^{i-v(*£](!*+»>}.- («> 



l=p(E/p)i{l^^rf a -±ty . . . (43) 



