444 Proceedings of Royal Society of Edinburgh. [sess. 
The consideration of the case when p — n leads to his second pro- 
position. The natural addendum is then made regarding the multi- 
plication of more than two determinants of the same degree 
(p. 310):- 
“ Datis quotcunque eiusdem gradus determinantibus, eorum 
productum ut eiusdem gradus exhiberi posse determinans, cuius 
elementa expressiones sint rationales integrae elementorum 
determinantium propositorum.” (xviii. 7) 
The equally natural transition to the subject of the multiplication 
of two determinants of different degrees results in the proposition 
(p. 311):- 
“ Sit pro indicis i valoribus 0, 1, 2, . . . . , m, 
c® = d(%W + a®af + . 
n n ’ 
pro indicis i valoribus maioribus quam m, 
_./*), (}) aQc) , (0 a W , 
c k~ a i + a i+ 1 i+i + a i+2 a i+2 + • • • 
n n 
erit 
2 + acq' . 
... a «2±aa 1 '....a#2± CCl V. 
M » 
' * * • C n ‘ 
Proposition IV. concerns the case where p >n. Proposition Y. is 
but a corollary to the combined propositions L, II., IV., its subject 
being the effect of the specialisation 
a k ~ a k * 
The enunciation is as follows (p. 312) 
“Posito 
C f= ( ®=«V + afaf + 
tC Z 11 
sit determinans 
= P; 
ubi p<n fit 
o 
II 
Plh 
ubi p — n fit 
P = 1 2 ± aa{ .... 
ubi^>>w fit 
P = S{2±o a' i 
' a m< n ) } ’ 
