426 An overlooked Discoverer in the Theory of Determinants, 



all the elements 1 whose column-number exceeds the row- 

 number by unity. This, of course, is a specialization more 

 apparent than real. Then, for the case of the fifth order, the 

 theorem is 



«! 1 



h b 2 i 



C\ c 2 c 3 1 

 d x d 2 d 5 d 4 1 

 e x e 2 e 3 e A e 5 



_ (a, b, c, d; e\ + fa, b, c, d; 

 1 VI; 2, 3,4, 5/i Vl ; 2, 3,4, 

 _/«, b,c,d; e\ +hd 

 VI; 2, 3, 4, 5/3 



= 2/yM h c,d;e\ m 

 ,-=o V ; VI; 2, 3,4, b); 

 and, quite generally, 



G) 1 ° 



(?) © 1 

 (?) ffl © 



5 A 



=lf-'>-(I; 2 -":. l! .i).' 



©$©©•■•© 



the number of terms on the right-hand side being evidently 2". 

 The theorem enables one to write out the final development of 

 a determinant of this kind currente calamo. Thus, to return 

 to the above determinant of the fifth order, we have at once 

 as its equivalent 



e x — {a& + We z + c^ 4 + d x e 5 ) -f- (aj) 2 e$ + a L c 2 e± + a 1 d 2 e 5 + l v c z e^ 



+ &i^5 + M^b) — («i^2^3^ + a x b 2 d 5 e 5 + a&d^ + b^d^) 

 + ajb 2 c z d±e h . 



Sect. I. Chap. 5. 



This ostensibly concerns the quotient of a determinant of 

 the order oo by another determinant of the same order, the 

 determinants being different in only one of their rows. The 

 following will suffice to show the nature of the single result 

 which is obtained. 



It is readily seen that, as a very special instance of the 

 theorem of chap. 2, 



I a x b 2 c^d 4 1 1 a 1 G 2 d z eJ h | — |<W^4 1 1 aj) 2 c % dj b \ 



+ \a 1 c 2 d 3 J\\\a l b 2 c h d i e d \ = 0. 



