EXPANSION PROBLEMS. 411 



Let p be restricted to the region To. Suppose first, for simplicity, 

 that the last two of the boundary conditions do not involve the 

 point 0, so that a^-i and av are zero. Then in each of the minors of 

 the last two rows of the determinant, an exponential factor can be 

 taken from each column, and the expansion according to these minors 

 has the form 



(40) Z5y/f,ye''^"'^+"^')Ml], 



where 5^;^/ and ^jjf are determinants of i' — 2 rows and 2 rows respec- 

 tively, with powers of the quantities lOt as constituents. Of these deter- 

 minants, 5i2, 5i3, ^n, and fi3 are surely different form zero, having 

 essentially the same form as 5i and 82 in the preceding section ; it is im- 

 portant to observe that when the roots Wj are arranged in cyclic order, Wi 

 and W2 are adjacent, and likewise Wi and W3. It is immediately recog- 

 nized that if p is in the remote part of To, all the terms of (40) excepting 

 possibly that with the subscripts 1, 3, are insignificant in comparison 

 with the term that has the subscripts 1, 2, and may be merged with 

 the latter term, by virtue of the latitude that is allowable in the 

 interpretation of the bracket symbols. We may write 



(41) Ai = [dioSioleP^"^' + W2)n^ [Sisrisle"^"" + '^^''. 



This is on the assumption that a^^i and a^ are zero. Among the terms 

 which have to be added to those already taken into account, if this 

 restriction is removed, the worst that can occur is e"^'^ multiplied 

 by a power of p, with a further factor which remains finite. Such a 

 term will be negligible in comparison with e''^^i + ^^'^, provided that a 

 power of p is negligible in comparison with e''^'^, which will be the case 

 if the vertex W2 of the polygon representing the I'oots remains at the 

 right of the axis of imaginaries after the polygon has been rotated 

 through an angle of t/v, that is, if 27r/V < ^tt, v > 4:. As the condi- 

 tion last stated was imposed at the beginning of the section, the 

 terms containing power's of p may be incorporated in the expression 

 [5i2fi2]e''^'^'+ ""'^^ and the validity of (41) is general. 



Suppose the bracket symbols in (41) replaced by their limiting 

 values, and the resulting expression set equal to zero. Since 



W2 — Ws = e" — e " = e " \e " — e " j , 



the equation so obtained, which obviously has only roots of the first 

 order, is equivalent to the relation 



(42) 2pTi e "sin — = log ( — r^^ ) -f- 2mri, 



V \ 612^12/ 



