And when the two factor lines are at right angles to one another, or 0=42, the product is simply ix" -}-jy"-}-**kz**", the unit line perpendicular to both.

Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i d x +j - + k dz could be, like ix+jy+ **kz**, effectively expressed by a single letter.

Suppose, therefore, that the equation of the boundary is r =a+a cos **kz**, (I) where a is a small quantity, the axis of z being that of symmetry.