926 
Proceedings of Royal Society of Edinburgh. [sess. 
definition, is “ Cramer’s rule ” ; but the first that is actually 
formulated and numbered is one of much later date than Cramer, 
“ Theorem I. — If the whole of a vertical or horizontal row 
be multiplied by the same quantity , the determinant is multi- 
plied by that quantity .” 
In this form, as is well known, it afterwards became almost 
stereotyped. The second result, which is of about the same age, 
is that regarding a determinant whose vertical row consists of 
^-termed expressions, second vertical row of ^-termed expressions, 
third vertical row of r-termed expressions, and so on, being to the 
effect that such a determinant is expressible as a sum of pqr . . . 
determinants with monomial constituents. The next seven results 
are, like the first, new only in form, the wording being, as in 
Theorem I., more topographical in character than formerly, on 
account of the determinant being now more consciously viewed as 
connected with a square holding n-n quantities situate in n vertical 
rows and at the same time in n horizontal rows. The tenth result, 
which is a converse of the ninth, is new but unimportant, viz. — 
“ Theorem X. — If a determinant of the n th order vanishes , a 
system of n homogeneous linear equations , the coefficients of 
which are the constituents of the given determinant , may 
always be established .” 
The eleventh result is the multiplication-theorem; and here 
anything that is noteworthy is not in the enunciation but in the 
proof. Beginning with two sets of equations, exactly after the 
manner of Joachimsthal, viz., 
he views them as six linear equations in x , y , z , u Y , u 2 , u 3 , and 
seeks to find the value of one of the first three unknowns, say x, in 
two different ways. Firstly, by using the first set to eliminate 
u x , u 2 , u 3 from the second set, he obtains 
+ If I + ^ 3 C 1 ) x d- (^1®2 d* If 2 d" l^C 2 )y + (lyJ's + tf% + )z = 
( m ^ + mf A + mycfx + (m 1 a 2 + mf 2 + m 3 c 2 )y + (m 1 a 3 + mf 3 + m 3 c B )z = v 2 
(fqtq + nf l + n 3 c x )x + (n x a 2 + nf 2 + n 3 c 2 )y + ( n Y a 3 + nf z + n 3 c 3 )z = n 3 
viz. 
