551] 
two smith’s prize dissertations. 
55a 
and it is 
< Q (# 4 + 8 a? -f 43# 2 + 24#) 
— P( 14# 3 + 33# 2 + 30#), 
the difference of these limits being 
= (Q — P) (# 4 + 22 # 3 + 7 5# 2 + 54#). 
2. A determinant is a function of n 2 letters ; viz. arranging these in the form of 
a square, the determinant 
fti} , Cj 
a 2 , 6 2 , c 2 
a s , 63, C3 
is a function linear in regard to each of the n 2 letters, and such that interchanging 
any two entire lines, or any two entire columns, the sign of the determinant is reversed, 
its absolute value being unaltered. 
The above definition leads to a rule for calculating the actual value of the deter 
minant, which rule may be taken as a definition, viz. the determinant is the sum of 
1.2.3...W terms obtained as follows: starting from the term 
+ ciib 2 c 3 ..., 
we permute in every possible way the suffixes 1, 2, 3, ... , and give to the term a 
sign, + , which is that compounded of as many — signs as there are cases in which an 
inferior number succeeds a superior number. Or, what is the same thing, any arrange 
ment may be obtained by a succession of interchanges of two letters; and then taking 
for each interchange the sign —, we obtain the sign + of the term in question. The 
positive and the negative terms are each of them £ (1.2.3 ... n) in number. 
To show the connexion of the two definitions, it is sufficient to observe that in 
the second definition, attending for instance only to the first and second columns, to 
any terms Maj6 2 , Na^l&c., there always correspond other terms - Ma 2 b 1} — ATa s b 1} &c., 
so that taking the pairs together, these are M (cqA — aA), N{a L b s — a 3 6 a ), &c., terms 
which change their sign, but remain unaltered as to their absolute values by the 
interchange of the first and second columns. 
Among the fundamental properties of determinants are as follows : 
The properties are the same as regards lines and columns. 
A determinant vanishes if any line vanishes (that is, if each term of the line is = 0). 
A determinant vanishes if two lines are identical. 
A determinant is a linear function of its lines. 
Whence— 
Determinant having a line sA is = s times the determinant having the line A (sA 
is here used to denote the line each term of which is s times the corresponding term 
of the line A). 
C. VIII. 
70