510 Proceedings of Royal Society of Edinburgh. [sess. 
— a theorem which degenerates into Cayley’s [Quart. Journ. of 
Math., ii. pp. 163-166) when <f> is put equal to 6. 
(6) Fourthly, all the theorems given in the paper referred to in 
§ 3 are capable of the same extension as Sylvester’s. Only one of 
them need be quoted in its generalised form, viz . : — If in the con- 
tinuant of the n th order 
6 b } 
fin- 1 4 > 
fin-2 0 \ 
fin-3 4* 
the difference between the element following any 6 and the element 
preceding the same be constant , equal to Iq say ; and the correspond- 
ing difference in the rows containing cf> be also constant , equal to fi 1 
say; then 
64> - b x fi Y 
is a factor of the continuant , the cofactor being the similar con- 
tinuant of the (n - 2) th order ichose minor diagonals are got from 
those of the original by striking out Iq , b 2 from the one and fi 1 , /3 2 
from the other. 
(V) 
n 
(7) Fifthly, with the notation of § 4 we find 
0 x - n + 2 
r <f) x — n + 3 
x - 1 0 x -n + 4 
x - 2 c/> 
n- 1 
1 
• (f % + 1 ) (x + 1 ) • cr n _2 
+ (^ ~^ ■ (x - n + l) (x + l) - (x - n + 2)x‘a n _ i 
(VI) 
where the putting of <£ = 0 gives a theorem first published in the 
paper referred to in § 1. 
