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Proceedings of the Eoyal Society of Edinburgh. [Sess. 
where for the purpose of verification we should of course use, not Laplace’s 
expansion-theorem, but Cayley’s. 
(7) A second point in advance is that the type of theorem which we 
have thus far been dealing with is not confined to the field of determinants, 
but finds a place also among Pfaffians, where there are analogues to both 
the theorems of §§ 5, 6. The latter analogue of the two is remarkably 
close. For example, taking the Pfaffian 
' 1 I ’ 
and the places occupiable by zeros to be in the axis of symmetry — that is 
to say, the places (1, 5), (2, 4), (3, 3) — we have the equality 
tto a. 
d, d. 
+ 
21 
d. d^ 
(8) A third step which has been found possible in the direction of 
generalisation is still more curious. In all the preceding paragraphs zero 
elements have taken a somewhat conspicuous place : we have now to learn 
that in the string of determinants forming the left-hand member of our 
main theorem (§ 5) no zero elements are called for at all, that, in fact, 
subject to a certain condition we may substitute for the zeros in these 
determinants any quantities whatever. This condition simply is that the 
zero occupying a definite fixed place shall always have the same substitute. 
For example, in the case previously taken where n = 5, r — 2, s = S, and 
where there are three determinants not free of zero elements, the substitute 
for a zero in the place (1,4), if in one instance made x, must in every other 
instance be made x : in short, the substitution must be of the nature 
(1.4) 2/. 
(2.4) (2,5) j U v' 
and our new result is that, this change having been performed on the left- 
hand member of the equality, the right-hand member is unaffected. In 
verification we have only to show that the additions thereby made to the 
