222 Proceedings of Roycd Society of Edinburgh. [ sess . 
steps by which any number p reaches the place it is intended 
finally to occupy in that permutation, that if p should advance 
in the first instance m places beyond this, it must subse- 
quently return through m places : or, which is the same thing, 
it must at a later period of the march, allow m of those which 
it has passed to repass it, so that it will regain its proper place 
after the number of steps has been increased from q to q + 2m, 
which, by the rule, require the same sign as q. The same 
reasoning applies to every other figure ; and hence the consis- 
tency of the rule is evident. (hi. 25) 
He then establishes four properties of the functions, viz. (1) 
Vandermonde’s theorem regarding the effect produced on the 
function by transposition of a pair of letters; (2) Bezout’s recur- 
rent law of formation ; (3) Scherk’s theorem regarding the partition 
of one of the functions into two ; and (4) Scherk’s theorem regard- 
ing the removal of a constant factor from one of the functions. 
The two latter theorems, which, as we have seen, had been stated 
for the first time only six years before, are given by Drinkwater in 
the following form (p. 27) : — 
(8) If any factor in /{XYZT. . . ( n ) }, as X, be divided 
into two parts, X = V + W, the function may be similarly 
divided, so that 
/{(V + W)YZT . . . (n)} =/{VYZT . . . (n)} +/( WYZT . . . (n)} 9 
placing each part of X in the same relative position (which in 
this example is the first) which X itself occupied before the 
division. (xlvii. 2) 
(9) If any quantity which does not vary from one equation 
to the other, and which, therefore, is not liable to be affected 
with an index, is found under the symbol, it may be con- 
sidered a constant coefficient of every term of the developed 
function : and written as such on the outside of the symbol : 
of this nature are the unknown quantities themselves, so that 
for instance, 
f{XYxZT .... (n)}=xf{XYZT . . . (»)}, 
and so of like quantities.” (xlviii. 2) 
After these preliminaries the problem of the solution of n linear 
