EXPANSION PROBLEMS. 415 



consisting of tlie principal terms in the expansion of the modified 

 determinant according to the minors of its first two rows; here ^'2 

 and o's are quantities having the same absolute value, different from 

 zero, and the constant factor just referred to is assumed to have been 

 determined so that they are conjugate imaginaries, and /c'2 is the order 

 of the highest derivative whose value at the point enters into the 

 second of the adjoint boundary conditions. Let us change our earlier 

 notation to the extent of representing the new characteristic function, 

 instead of (44), by rn(.r). The claim that this function is sufficiently 

 well represented by the function vn{x) is to be understood as follows: 

 Let/(.r) stand for the function (14), the integer q being left arbitrary 

 for the present. If the bracket symbols are given the more specific 

 interpretation explained in the latter part of the preceding section, 

 and the various terms are integrated by parts, it is found that 



^ prnr cot (2ir/v) 



(45) I rf(x)Vn(x)dx- rf(x)Vn{x)dx 



Jo Jo 



where c is independent of n, like the quantities to be denoted by 

 the same letter with a subscript in the remainder of this section. The 

 value of/f{x)r^(x) dx difl'ers by less than cie""^ ^ot {2./") / ^g + s from 



(46) -^, e""---^ 





5'3 



^ gPnWSTT 



Pn 



3+2 





in comparison with which expression the difference just mentioned and 

 the quantity (45) are insignificant, unless the four terms obtained by 

 expanding the two determinants nearly or quite destroy each other. It 

 is to be shown that this mutual destruction will not take place, at 

 least if q is suitably chosen. 



If the first determinant is divided by ic^"'^'-'^^'-, and the second 

 by ii'r^~-+*^'2, the remaining factors will be equal in the two cases, 

 except as to sign, since u\/wo is the same as li's/wi, each being equal 

 to g-^"/". The common factor e2(g + 2)«/. _ ^-2k',.i/. ^.j]! vanish 

 only if —q-2 is congruent to A- '2 (mod. v), that is, for only one of 

 any v consecutive values of q. On the other hand, the expression 



(47) -^;- U'^ W2-«-2+^'2 f"""^- + b'z Wi-^-2+^'== eP-'^A, 



which forms the remaining factor of (4G), can be treated as the expres- 

 sion in braces in (37) was treated; in any v consecutive values of q 

 there can not be more than one for which its absolute value fails to 



