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Proceedings of the Royal Society of Edinburgh. [Sess. 
which may profitably be compared with the result of § 8. Of the new 
forms of unisignant which make their appearance in it, the first and the 
last are of little importance, being really both included in the simple form 
• a l a 2 a 3 
ft * . . 
.ft • y • 
ft • • * 
which equals afiSpyz + a 2 /3 2 zx + affi z xy : but the form which includes the 
two others is not at all so trivial, the proposition regarding it being that 
any positive elements whatever may be inserted in the vacant places of 
the determinant 
2 111 
-a - a 
~P ' ~P 
-y 
-y > 
(XIII.) 
and yet all the terms of the final development remain positive. Further, 
if the elements so inserted be a 3 , a 2 , ft , ft , y 2 , y 1 , the determinant 
a + a 2 + a 3 a 2 a g 
ft A + ft + ft ft 
7l 72 7 + 71 + 72 
has the same development, namely, 
(XIV.) 
afiy + aft 7l + y 2 ) + fty(a 2 4- a 3 ) + ya(ft + ft) 
+ a (ft7l + ft7l + ft72) + ft a 27l + a 272 + a 372) + 7( a 2.ft + a 3ft + “sft) 
+ 2(«2ft7l + a 3^l7‘2 ) ? 
a result which ought to be classed along with Sylvester’s of the year 1855, 
as his determinant (which is also unisignant) differs from it only in having 
the non-diagonal elements all negative. 
12. From any one of the forms (VIII.), (IX.), (X.), we can readily 
verify the fact that the number of terms in the final development is sixty- 
four. This agrees with the result of putting each variable equal to 1 in 
either of the determinant forms, — a substitution which is warranted as 
soon as the unisignancy becomes established. When the determinant is 
of the n th order, the number of variables is 2 n_1 , and the number of terms 
in the final development is 2 (n-1)(n-2) . (XV.) 
13. We may note, in conclusion, that when in the determinant of § 1 
the function V is different in form from Boole’s, the result is different in 
