174 Proceedings of Royal Society of Edinburgh. [sess. 
0 ; ' 
We observe that the directed sine may be broken up into two 
components — namely, cos c sin 6. /3 + cos 6 sinc-y, which is per- 
pendicular to the axis of revolution, and -sin 6 sine sin /2y./3y, 
which has the direction of the negative of the axis of revolution, 
for /3y is identical with a. 
Draw OS to represent the first component cos c sin 1-/3, OT to 
represent the second component cos b sin c-y, and OU to represent 
the third component -cos& cose sin/lya Draw OV, the result- 
ant of the first two, and OR, the resultant of all three ; then 
cos a = cos b cos c — sin b sin c cos /3y 
^ t OR _ cosc sin b-(3 + cos b sin c-y - sin b sin c sin /3ya 
sin a \/l - (cos b cos c - sin b sin c cos (3y ) 2 . 
The plane of OA and OY passes through OR, which is normal 
to the plane POQ ; hence these planes cut orthogonally in a line 
OX, and the angle between OA and OX is equal to that between 
OY and OR, for OY is perpendicular to OA and OR to OX. Let 
6 denote the angle AOX ; then 
sin 0 = sin b sin c sin /3y 
J (cos b cos c — sin b sin c sin /3y) 2 . 
The figure (fig. 3) represents a section through the plane of OA 
and OY ; MX represents sin 0. Hence the axis £ can be put 
in the form cos 6-e - sin 6-a, where e denotes a unit axis per- 
pendicular to a. The unit axis e may be expressed in terms of two 
axes j and k, forming an orthogonal system with the axis of 
revolution, which may be denoted by i. Hence a perfectly 
general expression for any spherical versor is e a V _ Y, where 
£= f - l{cos 0*(cos <f>‘j + sin (f>‘k) - sin 6'i}. 
We observe that if e&v'-i'S is an angle in the double sheet, 
\/ - 1£ is a vector to the surface of the single sheet. 
It is now easy to find the solution of the analogous problem, 
namely, the product of two diplanar hyperbolic versors when the 
plane of each passes through the axis of revolution. 
The axis of the versor is perpendicular to the plane of the versor 
when the latter passes through the axis of revolution ; and we shall 
assume that it is of unit length, an assumption which is afterwards 
