2^2 Report S.A.A. Advancement of Science. 



6. The general theorem which includes Hunyady's case 

 regarding Aio, Brill's and Scholtz's regarding Ae, and the still 

 simpler case 



A3= ni^ a^ 2a^a2 = |«l62|^ 



is formally enunciated by Pascal in his text-book "I Determinanti," 

 published at Milan in 1897 {see pp. 134-137). A separate section 

 (§ 26) is devoted to it ; but, strange to say, the heading of the 

 section is "Teorema di Hunyady," a name which apparently* he 

 has taken over from Igel and Escherich, but which in view of what 

 precedes he will surely reconsider. 



7. Be this as it may, let it now be noted that if the point of 

 view in regard to the theorem be entirely changed, so that we look 

 upon the determinants in question as eliiiiiiianis, a striking advantage 

 is gained, the following general theorem being at once reached : — 



// a set of n homogeneous equations of the first degree in n 

 unknowns be given, the determinant of the set being A, and there be 

 formed another set consisting of all equations of the p^^ degree derivable 

 from the equations of the given set bx niulliplication among themselves, 

 the determinant of the latter set is equal to 



.Cfi + n-l. n 



By way of proof we note in the first place that the number 

 specifying the order of the derived determinant is the number of 

 arrangements of n things taken p at a time with repetitions allowed^ 

 and therefore is Cn+p-i,p : secondly, that the degree of each element 

 of this determinant is p : and consequently that the degree of each 

 term of the finally expanded determinant is 



p . Cn^p-ip . 



Further we note that on account of the mode in which the second 

 set of equations is derived from the original set, the eliminant of the 

 second set cannot contain any factor extraneous to the eliminant of 

 the first set nor give prominence to any factor of the first set over 

 any other, and consequently that the former eliminant must be an 

 integral power of the latter. As the degree of each term of the 

 latter is n and of the former p . Cn+p-i,p, the index of the power 

 referred to must therefore be 



P C 



~ . *-'?/+ p-i,p 



n '^ 



I.e. — . C-7J j.p_i »i_i 

 n '^ 



I.e. (^i,+p-l,n, 



as was to be shown. 



The theorem used in the Phil. ]\Iag. for October, 1902, and 

 formulated by Pascal in his text-book is the case of this where p=2. 



* Their papers in the Monatshefte f. Math. u. Phys. III. pp. 55-67, 68-80, 

 I have not been able to see. 



