50 THOMAS MUIR ON GENERAL THEOREMS ON DETERMINANTS. 
corresponding element of the (g+1)™ row, each element of the (p+1) row 
from the corresponding element of the (q¢+ 2) row, and so on until the elements 
of the g row have been reached and subtracted, we have 
yy yg Cya.o yen Gp este ia 9. ung, a eae nia 
Bn. ling: Cag spe biel Conoger Coy 0 epee ee ll 
Gg, Ago Ugg . +. Agp « As, O 0 (zn 
| Ap, App Ups yp An, Y 0 lyn 
ges 
gy Con (araed dle becurs Geko «ists RAO spon. hen 
OO Ore eaten eMac Ok, Upp 2B ncaa atl’) 
Diese 0 Reta tO te enOP Win. cm sll ety ee 
Cy. Onde Onna tnt h Cnnte won Uigljp incua Onan teams Oy, 
But here there is a set of g—p+1 rows having only one non-zero minor of 
the (y—p+1)™ degree, and this minor being 6 and its complementary D(q,,) 
we have 
AO DG), 
the sign of the product 6D(q,,) ‘being positive, because the numbers of the 
rows in which the elements of 6 occur are the same as the numbers of the 
columns, and the sum of the two sets of numbers therefore even. This identity 
is what remained to be proved, hence the theorem is established. 
In the introductory example all the rows but two were overlapped, the chosen 
minor of D(a,,) being thus of the third order. The other possible cases for the 
same determinant are (2) where the chosen minor is of the second order, 
say | 435 @, |, and then we have 
| 39 44D (Ay5) = 441% o9b 39% 44l|4g9% 44455] — [Cy 1 4o3%34@45||%39%43%54] + [112% 03%g4%45||%g1%43%54 | 
(3) where the chosen minor is of the first order, a, say, in which case we 
have two identities, viz., 
tsgD (dys) = | @y1 gb || 33% 44455 |—| 11 %03%3¢ || C3%43%55 [+ | 11% 93%95 || 0% 4354 | 
+ | Ayo%o3434 || C31 %43%55 |—| A12%3%35 || 31% 43%54 | +] @13%24%3p5 || 31% 49%ss | » 
and 
A330) (yz) = | Agy@gg || go@43454%15 |—| C0439 || @31%43%s4%15 |+| Cos%34 || 3144245315 
— | Ay3g5, || Mp1 Ugo%53%4 | 
