140 LOTKA. 



The law of expansion of the determinant A (X) is readily recognized 

 if we use Cauchy's notation, in which a determinant is defined by 

 writing down the terms of its dexter diagonal enclosed in parentheses. 

 In the case that n = 4, for example, we have^ 



X"* — {flu + fl22 + fl33 + fl44}X^ 



+ {(«iia22) + (ffiiflss) + (flu an) + (022033) + (022044) + (033044)1x2 



— {(011022033) + (011022044) + (O11O33O44) + (a22 033 044)}X 



+ (flu 0-22033044) = (10) 



To determine the coefficients a', a" .... we substitute in (7) 



' o^t 



.Ti = ai e 



.V. = ar e^'-' \ (11) 



Equating coefficients of homologous terms we then obtain 



a'i (oii — X,) + a'i fli2 + a'i' a^ + . . . = 



Oii O2I + Oi'i (022 — X,) + a'i' fl23 + . . . = 



a'i flsi + a"i fl.32 + Oi'i (fl33— Xi)+ . . . = 



(12) 



The equations (12) are not independent, seeing that A (X) vanishes 

 according to (9). 

 The trivial solution (13) 



a'i = a"i = a'i' = ... =0 (13) 



of (12) is of no interest. The existence of other solutions is assured 

 by the vanishing of A (X) according to (9). One of the constants 

 a'i, a'i,... remains arbitrary, and the proportion a i: a'i: a'i':. . .is 

 then determined in the well-known manner by the homogeneous 

 linear system of equations (12). That is to say, if we form the 

 determinant of the coefficients of a'i,a"i, a'i',. . . in the left-hand mem- 

 bers of (12), and if we denote bv Aa', A^ , Aa'", . . . the minors of the 



coefficients of a ', a'i, a"/ ... in any one selected row, then 



a'i:a"i:a':': : : A„/: A«^: A^';/: (14) 



6 See, for example, Czuber, Einfiihrung in die hohere Mathematik, p. 178. 

 The symbols enclosed in round parentheses denote the determinants of which 

 these symbols form the dexter diagonal . 



