210 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XI. — The Product of the Primary Minors of an n-\yy-{n+\) Array. 
By Thomas Muir, LL.D. 
(MS. received October 7, 1907. Read November 4, 1907.) 
(1) In a paper by A. Scholtz entitled “ Sechs Punkte eines Kegelschnittes ” 
( Archiv d. Math . u. Phys., lxii. pp. 317-324, year 1878) there appears a 
statement which, after correction of two misprints, runs thus : — 
VlVn 
VlZn + l/nZl 
VnVi 
VnZi +ViZn 
ViVl 
Zi*l 
Vi*l +Vl*i 
where = y l z n — y n z u A year or so later there was published a paper by 
Hunyady with the title “ Beitrag zur Theorie des Flachen zweiten Grades ” 
( Crelles Journ., lxxxix. pp. 47-69), in which it is asserted, again without 
proof, that 
VlV2 
Z l Z 2 
PlP2 
V\ z i +V&. 
2 /lP 2 + 2 / 2 Pl 
hPv + hPi 1 
ViVs 
Z 1 Z 3 
PlP3 
yi z 3+yz z \ 
V\P?. "t UzPi 
z iP e + z sPi 
ViV* 
Z 1 Z 4 
PlP4 
yfr+yfi 
2/1P4 + 2/4P1 
Z1P4+Z4P1 
2/ 2 2/ 3 
Z 2 Z 3 
P2P3 
y^s+Vzh 
2/2P3 + 2/3P2 
hPs + hPz 
2 / 22/4 
Z 2 Z 4 
P2V4 
VA + Vih 
I/2P4 + I/4P2 
hP t + ?-iP<t 
2/32/4 
Z 3 Z 4 
P2V4 
VA + Vih 
V3P4 + V4P3 
hPi + hPz 
^ 234 ^ 341 ^ 412^123 * 
The object of the present note is to formulate and prove a general theorem 
of which these are cases, and to draw certain deductions therefrom. 
(2) The theorem somewhat imperfectly enunciated is — The product of the 
n -line determinants formable from an array of n rows and n + 1 columns 
is expressible as a determinant of the order Jn(n-f-l). By way of proof let 
us consider the case where n = 4, and where, therefore, the array may be 
written 
CL i CL 2 CL^ 
b l by b z b 4 b 5 
C 1 C 2 C 3 C 4 C 5 
d i dy d z d 4 d 5 . 
The determinant of the 10th order which has to be proved equal to the 
product 
I | • j ci-fyCod b | • | CL-fyCjjd^ | • | a l h z c 4 d b j • j (Xyb^c^d b 
