THOMAS MUIR ON GENERAL THEOREMS ON DETERMINANTS. 49 
being the same as those from the 1* to the g™ in the given determinant, except 
that g—p+1 zeros are inserted after the g* element in each, and those from 
the (g+1)" to the last inclusive being the same as those from the p™ to 
the last in the original determinant, except that g—p+1 zeros are inserted 
before the p** element in each. Of this determinant the selected minor 6 
occurs twice as a minor, having zero elements below it in the one case and zero 
elements above it in the other. Hence, if we take the first g rows and form 
every minor of the g™ degree preparatory to finding the expansion of A as a 
sum of products of complementary minors, we see that, although the full list 
of minors would be exactly the same as if we had been dealing with D(q,,) 
instead of A, still we need only take those which include the selected minor 64, 
because all the others have here complementaries which vanish ; also we see 
that the complementaries of those thus taken are not of the degree n—g as 
in D(a), but of the degree »—p +1, each one including, in fact, the correspond- 
ing complementary in D(q,,) and the selected minor besides. Now it is evident 
that the sum of products thus found as the equivalent of A is exactly the sum 
of products referred to in the theorem, the addition of g—p+1 in the deter- 
mination of the sign of a product being due to the g—p+1 zeros which are 
inserted in A, and which for certain elements make the number of their column 
greater by g—p+1. It thus remains to show that 
A=D(dp) x8. 
Adding each element of the (g+1)* column to the corresponding element of 
the p™ column, each element of the (g+2)* column to the corresponding ele- 
ment of the (+1), and so on, as far as the zeros continue, we have 
Oi, M12, Org Cail ay, O 0 Cn 
Oe yp oR Com: og O 0 Can 
31 Az9 Ass 3p sq O 0 C3n, 
Ap, Ape Ang App Upq 9 0 Opn 
ne i a eR Ma Sina fete 
Aq Uq2 A%Uq3 Aap - qq O 0 Gan 
Ap, Apg Upg Upp - Ung Upp pq pn 
Ag U2 Ags Cop - gq Cop gg Can 
Sige, Chie Cay” 3-2 Oia OC SR ERR 
Now, in this determinant, subtracting each element of the p‘ row from the 
