of Edinburgh , Session 1885—86. 
825 
The method of procedure is evidently quite independent of the 
order of the determinant, so that we have the general theorem 
(p+ iy 
(p + 2)' 
(P + 3)'.. 
. . (p + m- 
-2)' 
(2+1)' 
(2 + 2)' 
(g + 3)' . . 
. . (q+ m- 
-2)' 
(r + 1 )' 
(r + 2)' 
(r.+ 3>' . . 
, . (r + m - 
-2)' 
0 + 1)' 
( z + 2)' 
( z + 3)' • . 
. . ( z + m - 
-2)' 
i 
a 
a? 
a 3 
a m ~ 2 
(p) 
(p+0 
(p + 2) 
0 + 3).. 
. (p + m - 2) 
(2) 
(2+1) 
(2 + 2) 
(2 + 3) . . 
. (g + m-2) 
— 
(r) 
O + l) 
0 + 2) 
0 + 3).. 
. (r + m - 2) 
(*) 
0 + 1) 
0 + 2) 
0 + 3) • • 
. (z + m - 2) 
the number of letters, p, q, r, . . . 
, z being m 
-2. 
It should be observed that, as the demonstration does not take 
into account any properties of h n or h' m but merely the equation of 
relationship, this result is true of any functions which happen to be 
related as this equation indicates. 
3. The connection which the theorem Ji n = h n -ah n _ l9 audits 
generalisation have with the theory of alternants, arises from the 
fact that every h is representable as the quotient of two alternants, 
viz., the quotient of an alternant of the second simplest form, e.g. 9 
a a 1 
b 
c 
b 2 A 3 
c 2 c 3 
1 d d 2 d* d x 
1 e e 2 e s e x 
or [ cdb J c 2 d s e x 
(sc<4) 
by an alternant of the simplest form, e.g. f 
| a°b 1 c 2 d V | ; 
which latter being the difference-product of a , b, c, d, e we denote 
also by £*( abode ). In the particular case here instanced the 
suffix of A would x - 4. 
