656 Proceedings of Royal Society of Edinburgh. [sess. 
The result desired, viz. 
n (P r + Q r) _ rp 
P r + rcQ r ~ ’ 
is thus realised. 
In a subsequent part of the paper it is stated that in a quite 
similar manner the following more general theorem may be proved, 
viz., that if 
and 
S = 
6 l f 0 
d + b „ 
e ° + d + b 
b^c A 
c, + 
b r c r - 1 
d + b r+1 - c r _! 
T = 
^2 C 0 
d + bc 
b r c r - 2 
d+b r — c r _ x + 
d + h 
then 
(c Q <i)S + b^Q ^ 
S + 5, -i - 
Strictly speaking, this is the only result of Bauer’s which con- 
cerns determinants, and the rest of his paper might Avith justice be 
passed over. The main application, however, is so easily deduced 
therefrom, and is so interesting, that space may well be given it. 
There is, indeed, little more to be done than to combine the case 
Avhere Ci=b +1 with the case Avhere Ci=b i+2 . For the one substi- 
tution gives us 
(5 1 -5)P + 5 1 2 Q_5 1 5 2 
P + ^Q d + 
b r _,b r , , 
p v ’ + ~T~ +^’v i 
~ "7 7 T d + O r <-i 
Q d + & 2 — + 
- : 
d 4- 5 r+1 — b 
if 
