1907-8.] Dr Muir ou General Determinants. 693 
In reference to determinants with binomial elements (§ 13) he says : 
“ Compiendo questo sviluppo si ottiene la formula 
I a i 4 a l ^2 4 @2 C 2> 4 73 I = I a i r 3 I 4 I a \ ^2 73 I 4 I a i @2 C 3 I 4 | % ft 2 73 ! 
+ I a l C 3 I + I a l G 73 i + I a l ft C 3 I + I a l ^2 73 l 
che e facile da tenersi a memoria per la sua perfetta analogia collo sviluppo 
del prodotto di tre binomii.” 
After giving a sufficient condition for the vanishing of a determinant, 
he enunciates (§15) the converse, namely, When a determinant vanishes, one 
of the rows is equal to a sum of multiples of the other rows, basing its 
validity on the fact that the multipliers referred to can actually be found 
by solving a set of simultaneous linear equations. 
The multiplication-theorem for determinants , A 2 of the third order 
he seeks to establish (§ 31) by partitioning the product-determinant into 
twenty-seven determinants, and showing that the sum of the six which do 
not vanish is A X A 2 . 
Chios theorem of 1853 is introduced (§ 38) by noting that the resultant of 
a r x+b r y + c r = 0 (r = lt 2 , 3 ) 
may be viewed as the resultant of 
| a i \ I y + I a i c 2 1 = 0 ) 
I h I y + I a i c 8 D = o j , 
and that therefore 
I j a . I I a, c 2 I | 
, . “ , | must be a multiple of I a, b 0 Co I . 
I G I I ^3 I I '12 3, 
That it is so he proves by diminishing the 2nd and 3rd columns of | a 1 b 2 c s \ 
by bJa-L times the 1st column and cja 1 times the 1st column respectively. 
Further, he points out (§§ 39, 40) a practical application, namely, in evaluat- 
ing a determinant whose elements are given in figures. 
The adjugate determinant (unfortunately renamed associato) is dealt 
with (§§ 55-58) in connection with the solution of a set of simultaneous 
linear equations, the special cases being considered where the determinant 
of the set is 1 and 0. In the former special case he notes the theorem, 
The adjugate of the product of two unit determinants is identical in all 
its elements with the product of the adjugates of the said determinants ; 
and in the latter the theorem all but reached by Jacobi in 1835 and 1841, 
In a zero determinant the cofactors of the elements of a row are pro- 
portional to the cofactors of the elements of any other row. 
Cauchy’s “ clefs algebriques ” ( chiavi algebriche) are expounded at some 
length (§§ 81-88). 
