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SYLVESTEE'S AND OTHEE UNISIGNANTS. 
By Thomas Mum, LL.D., F.E.S. 
(Eead March 15, 1911.) 
1. The first to draw attention to the existence of determinants having 
their final development free of negative terms appears to have been 
Sylvester. In the Quarterly Journal of Mathematics, i. (1855), pp. 42-56, 
when dealing with the subject of the change of independent variables in 
certain differential expressions, he was brought up against an n-line 
determinant of the type 
which, after some investigation, he asserted to contain in the final form 
of its development nothing but positive terms. No proof was adduced of 
the assertion ; and, save that he gave a rule-of-thumb for obtaining the 
development referred to, no light was thrown on the peculiar constitution 
of the determinant. 
2. Evidently it may be defined with reasonable minuteness as the 
determinant which has its non-diagonal elements all negative, hut the sum 
of the elements of each roio positive. Note may also be taken that the 
number of variables is the same as in a general determinant, but that 
those associated with the diagonal have the peculiarity of being equal 
to the sums of the rows. Another ready deduction from the definition is 
that everij co-axial minor of such a determinant is a determinant of the same 
hind. (I.) 
3. The property of unisignancy is most easily established by making 
the ^^th case depend on the [n - l)th and lower cases, and by the use 
of Cayley's development proceeding according to products of the variables 
special to the diagonal. Thus, on the supposition that the cases for the 
second and third orders have been established, we take the case for 
the fourth order, the determinant then being 
s a + a^ + -\-a^ - a^ - a^ - a^ I 
a + ^2 + a^ 
- &i 
b-hb^ + b 
-c^ 
- a^ 
c + c^^c^ , 
C + C^+C^ + C^ 
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