824 
Proceedings of the Poyol Society 
2. This theorem admits of a wide generalisation, the direction 
of which may be suggested by writing the right-hand member as 
a determinant, so that the simplest case takes the form 
1 a I 
or, say, rather 
h'p+i — 
h„ 
h. 
h 
'p+i 
(p+iy= 
which again, merely for shortness’ sake and in order to give greater 
prominence to the suffixes, may be written 
1 a 
(p) (p+i) 
To take the next case in order of complexity, we have 
(p + 1)' ( p + 2)' = (p + 1) - a(p), (p + 2) - ci(p + 1) 
(2 + 1)' (? + 2 )' (g + l)-a(?), (q + 2) - 0(2 + 1) 
(p + 1) <P+ 2 ) _ (f) (P + 2 ) , (p) (P + 1) 
(2) (2 + 2 ) 
la a 2 
(p) (p+ 1) (p+ 2 ) 
( 2 ) (2 + 1 ) (2 + 2 ) 
(2+1) (g + 2 ) 
+ a‘ 
(q) (2+1) 
In like manner we have 
■+a< 
(p+ 1) 
(2 + 1 ) 
(r + 1) 
(P + 1) 
(2 + 1 ) 
(r + 1) 
(p) 
( 2 ) 
(r) 
(p+1)' (p + 2V (p + 3)' 
(2 + 1)' (2 + 2 )' (2 + 3)' 
(r + 1)' (r + 2)' (r+3)' 
(p + 2)-a(p+ 1), (p + 3) - 
«(?)» (2 + 2)-«(2 + l)» 
a(r), (r + 2) - «(r + 1), 
(P + 2) (p + 3) (p) 
(2 + 2 ) (2+3) ( 2 ) 
(r + 2) (r + 3) (r) 
(P + 1) (P + 3 ) (p) 
(2 + 1) (2 + 3) -a 3 ( 2 ) 
(r + 1) (r + 3) (r) 
la a? cfi 
(p) (p + 1) (P + 2 ) (p + 3) 
(2) (2 + 1) (2 + 2 ) (2 + 3) 
(r) (r + 1) (r + 2) (r + 3) 
(2 + 3)- 
(r + 3)- 
(P + 2 ) 
(2 + 2 ) 
(r + 2) 
(P+1) 
(2 + 1 ) 
(r + 1) 
a(p + 2) 
a(2 + 2 ) 
a(r + 2) 
(p + 3) 
(2 + 3) 
(r + 3) 
(P+2) 
(2 + 2 ) 
(r + 2) 
