472 Proceedings of Royal Society of Edinburgh, [july 16 , 
The assumption of the law may, however, he readily verified in any 
case, as follows — 
(1) Let ^ = %h^e^d^e^pW , 
which 
then 
11 11 11 . 
[^[3 |3_ 
(l + |2 + 3)= 
|2 |3 
2 |3 
(2) Let A = ... 
in which p indices are equal to a, q equal to /S, and the remainder 
r unlike. Expanding % as before, we get 
/ ■,^P+y+r-l ( I \p + q + r-\ |/) + g + r-l^ 
= (-l)^+*+’'.l^T^=j=i. {p + q + r) 
=^( - ]')?+«+»• 1 -^'*'^'*'^ 
- ■ Iz lx ■ - 
The following mode of regarding the problem may be appropri- 
ately noticed in connection with the foregoing. 
An examination of the method of expanding the function denoted 
by A, and of deducing the corresponding value of K, shows us that 
the process is exactly analogous to that of finding the number of 
permutations of any number of things taken all together. This 
method is precisely the same when the indices in a term of '% are of 
the most general character as when they are all unlike. To 
determine K in the former case, we have thus only to find the 
number of permutations of any number of things taken all together 
which are not all dilferent — a well-known algebraical proposition. 
Hence we have the rule that, if N be the whole number of indices 
or factors in a term of of which p are equal to a, q equal to P, 
r equal to y, &c., the numerical value of K is 
lA 
\p \q \r . . 
