(ii.) The elements of the theory of numbers belong to arithmetic. In particular, the theorem that if n is a factor of a and of b it is also a factor of pa= **qb**, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions.

For immiscible liquids the vapour pressure curve is the horizontal line ab, described so that aP = **QB** and bQ=AP. For partially miscible liquids the curve is Pa i b i Q.

The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of pa-**qb**, where p and q are any integers.