134 KANSAS UNIVERSITY QUARTERLY. 



allel rows with elements proportional; then J^o; and since we know, 

 (see Muir), that J=rD^~i, we must have the result D=o. 



The reciprocal may be demonstrated in the following manner: 

 suppose 



Bi=kBi, B2=kB2, .... ,Bn=kBii: then 



q ' p' q p' ' q p' 



D=aiBi+a2B2+ .... +a^Bii=k(aiBi +a3B2+ . . . -U si^B^)=o 



qq'qq' 'qq ^qp'qp' 'qP"^ 



II. 



Definition. — Every determinant whose elements verify the rela- 

 tion apq-faqp^^o, is called a symmetrical gauche determinant, when 

 the elements of the principal diagonal are all zero. 



Problem. — 'To show (i) that the value of every symmetrical gauche 

 determina7it of a?i uneven degree is zero; (2) that every symmetrical 

 gauche determinant of even degree is the square of a rational and entire 

 fiinction of the elements of the determinant. 



SOLUTION. . . 



(i). If the gauche determinant is of an uneven or odd degree, with 

 every term such as 



^ ^ B y k 



in which I and I' are the numbers of inversions of two sets of indices, 

 we may associate the term 



(— i)i'-riaBay, . 



\ J B' y 



whose values are at once zero, or are equal and have contrary signs; 

 their sum is, therefore, equal to zero, as is also the value of the 

 determinant. 



For the second part of the problem, let us begin with the following 

 identity: 



d^D _ dD dD dD dD 

 da^~das ~ da^ da^ "~ da« "da^ 



m q m q m q 



T^ J • • , dD 



D designating the determinant we are considering; . ^ its derivative 



m 



d^D 



with regard to a*" , and 1- ,. . ^ its second derivative with regard to a"^" 

 ^ 1^ da^ da'^ ^ m 



m q 



first, and afterward to a^. 



q • 



Let the degree of D be represented by n, and suppose 7n=^r=r^n, and 

 q^=s^n — I, the identity with which we started will then become 

 d^D dD dD dD dD 



(I) D 



daMa*^-} da" da^i-i da^^-^ da^^-i ' 

 n n— 1 n n— 1 n n 



We may show at once that D is a perfect square by putting, for 

 e?cample, ??=2, thus: 



o a 



-a 



P.- 



=r:a3 



