517 
of Edinburgh, Session 1865 - 66 . 
The two latter bodies being respectively the oxide and the sulphide 
of the former, as their formation and decomposition indicate. It 
would be easy to give a much larger number of examples, but 
these are sufficient to show the way in which this view of the 
atomicity of sulphur may be applied to explain the constitution 
of its compounds., 
4. Note ou a paper by Balfour Stewart, Esq., in the Trans- 
actions of the Koyal Society of Edinburgh, by I. Tod- 
hunter, Esq., M.A., St John's College, Cambridge. Com- 
municated by Professor Tait. 
In Volume XXI. of the Transactions of the Koyal Society of 
Edinburgh, pages 407-409, a proposition in the Theory of Numbers 
is demonstrated. The proposition may be extended, and, at the 
same time, the demonstration simplified. I propose to establish 
the following result : — Let 1, a, /S, y, . . . (x, he the n roots of 
the equation 
x^-l = 0 (1.) 
then, will 
= (2.) 
where t is any quantity. 
It is obvious that the term independent of t on the left-hand 
side of (2) is unity. Consider any other term, — for example, that 
involving f ; the coefficient of this is equal to the sum of the pro- 
ducts of every three of the quantities 1, a, y, . . . //,, with 
the sign changed ; and from (1) we know that this sum is zero. 
In this way (2) is established. 
It is known that if be a prime number, the quantities a, /?, 
y, . . . fx, may be expressed as the powers of any one of them ; 
for example, as a, a?, o? . . And if n be not a prime 
number, they may be expressed as powers of some of them. 
Divide both sides of (2.) by 1 - ^ ; thus — 
(1 — aC (1 “ ^0 (1- ~ yC • • • (1 — ~ ^ "h ^ *b + . . . 
In this result, suppose ^ = 1 ; then we obtain 
(1 -a) (1-./5) (1 -y) . . (l-/^) = /i. 
