554 two smith’s prize dissertations. [551 
Determinant having a line A + A' — determinant with line A 4- determinant with 
line A'. 
It follows that, if any line of a determinant is the sum of the other lines, each 
multiplied by an arbitrary coefficient, or, what is the same thing, if we can with any 
of the lines, each multiplied by an arbitrary coefficient, compose a line 0, then the 
determinant is = 0. 
The same principle leads to a theorem for the product of two determinants of the 
same order n, viz. it is found that the product is a determinant of the same order n, 
each term thereof being a sum of the products of the terms of a line of one of the 
factors into the corresponding terms of a line of the other factor. Starting with this 
expression of the product, we decompose it into a series of determinants each of which 
is either =0, or it is a product of a single term of the one factor into the other 
factor, and the sum of all these products is equal to the product of the two factors. 
If we have n quantities x, y,... connected by as many linear equations 
then the determinant 
a^x + b x y + g x z + ... = 0, 
a i> b 1} ,... 
ci 2 , b-2, c 2 ,... 
^3> K Gg, ... 
is = 0 ; 
and so, if we have n linear equations 
CLjX + b-^y -J- CjZ + ... = u, 
then each of the quantities x, y, z,... is given as the quotient of two determinants, 
the denominator being in each case 
cii, bi, Ci,... , 
^2) b 2 , c 2 ,... 
a 3> b s , G 3 , ... 
and the numerators being (save as to their signs) that for x 
ih, b u Ci,... , 
u 2 , b 2% c 2 , ... 
Uz, b s , c 3 , ... 
and the like for y, z,.... 
A determinant remains unaltered when the lines and columns are interchanged, 
the dexter diagonal (\) remaining unaltered.