As will be seen later, the fundamental i, j, k of quarternions, with their **reciprocals**, furnish a set of six quantities which satisfy the conditions imposed by Servois.

(Note that the z here occurring is only required to ensure harmony with tri-quaternions of which our present biquaternions, as also octonions, are particular cases.) The point whose position vector is Vrq i is on the axis and may be called the centre of the bi-quaternion; it is the centre of a sphere of radius Srq i with reference to which the point and plane are in the proper quaternion sense polar **reciprocals**, that is, the position vector of the point relative to the centre is Srg i.

If the refractive index for one colour be n, and for another and the powers, or **reciprocals** of the focal lengths, be 4) and 4)+4, then (I) dï¿½/ 4) = dn/ (n - I) =1 /v; do is called the dispersion, and v the dispersive power of the glass.

The polar form is {(u+p) cos 26} a+{(u-p) sin 20) a = (2k)t, where p and k are the **reciprocals** of c and a, and u the reciprocal of the radius vector of any point on the caustic. When c =a or = oo the curve reduces to the cardioid or the two cusped epicycloid previously discussed.

In The Case Of Negative Numbers And **Reciprocals**, Only One Is Involved; And There Might Be Three Or More, As In The Case Of A Number Expressed By (A B)".

We may also note that of the Archimedean solids: the truncated tetrahedron, truncated cube, and truncated dodecahedron, are the **reciprocals** of the crystal forms triakistetrahedron, triakisoctahedron and triakisicosahedron.