30 



Proceedings of the Royal Irish Academy. 



These expressions are respectively 



{MiiC'v + kuc' - Xug)y\ + Mu(gv + c')vi + (M^^c'v + kad'--\ng)yi+Mii{gv^c'')Vi+ . . . 



(44) 



{M.iC^v + hiC- - Xiiff) yi 4 M2,(gv + c'')v, ^(Mizc'v + hic' - \i2g)y2 +Mii{gv + c')?;, + . .., 



(45) 

 etc. They are the analogues not of (13), etc., but of those expressions each 

 multiplied by c. They are to be used instead of (13) et seq. in forming the 

 determinants „ni, olio, etc. 



The solution of the problem is contained in the typical equations 



i, 



<j, = (47rt)-' 



Ci'dfi 



-2;.„n,(vO(e^<^-**F6(.„-M) -6-'^(-*'i^j(.-„/.)l j 



+ a term derived from this by interchanging v„ vz ', 



(46) 



„n = (47rt) 



e'-^'dju. 



' {e^^"-'F^{v„ - m) - e-^("-''>Ft(v,,fx)} \v,xl^{u) - x(m)| du + 2^ jUi m\ 



a } 



- 2^ „n. (.',) (e^'^-'^i^iCi'i, - /.) - e-^'^-'^-fjCf,, lii)}^] 4 Denominator of (46) 

 + a term derived from this by interchanging v,, vj ; (47) 



6y; = -(47r0-' 



C^'dfx 



{«M(»-"'i?'„(i;„ - /x) - C>'«-«)i?'„(v„/ii)} {v.i/-(m) - xOOlf^w - 2^„n, (l^,) 



- 2/ijni, (.',)( (!^(^>i^„(v, -ju) - e-^<^-'''i''„(i'„iu)U6| ^ Denommator of (46) 

 + a term derived from this by interchanging v,, vz. (48) 



The contour is again an infinite closed one surrounding the origin and 

 which, of course, avoids the zeros of the denominator. 



In view of the length of this paper it seems unnecessary to give the 

 verification. One point to be borne in mind in it is that gv^ 4 cMs a small 

 quantity of order yu,"l 



If we prefer it, the two terms in (46), (47), or (48), may be combined into 

 one integral taken along a continuous path. In the integrand in the first 

 term substitute for vi, v-. in terms of fj, ; then let either value of the radical 



