316 Proceedings of Royal Society of Edinburgh. [sess. 
We easily obtain the following expressions, 
P = 2-Vr,- 
^■S(rV Yr-r) 
7r ~ ' 2 ' 2-T 2 Vr " ’ 
* '2fp-p)YrYr 
6 - 2-T 2 Vr 
The symbol M is obviously not distributive. 
3. We may proceed to resolve a motor in any direction by the 
usual method of resolution. Thus we may decompose it into three 
rectangular components having the same pitch. Any one of these 
may be regarded as the component of the given motor, and will 
correspond to the actual motion when the other components are 
regarded as being balanced by constraints. 
The motor 
ml = Sr'.uw + TV/-UY/ , 
= (1+7T)W, 
= - (1 + 7r)S V/TJ Vr-U V r - (1 + 7r)YYr' U V/’-U VY , 
represents in this way a motor - (1 + ir)SVr'UVr-TJYr, in the direc- 
tion of Vr, and whose vector displacement is -7rYYr'UYr, pro- 
vided that the rotation YYr'UYr-UYr is prevented by constraints. 
Thus the quaternion r is the symbol of a complex of motors, each 
of which is determinate when the direction of its axis is given. 
The complex is formed by the generators of a hyperboloid, whose 
axis is Yr, when the angle between Y r and Y r is fixed. The 
screws corresponding to the one set of generators have the opposite 
pitch to that of the other set. Each quaternion thus represents 
pairs of reciprocal screws passing through each point of space. 
And a set of three non-coplanar quaternions furnish a set of six 
reference screws through each point of space. By proper choice of 
the origin that set might be made the canonical co-reciprocals for 
any one point in space. By taking the three rectangular coordinate 
screws with both positive and negative pitches we get a set of six 
canonical co-reciprocals, to which any screw may be referred. 
4. In the case of a rigid body moving with two degrees of free- 
dom, the motion may be represented in terms of two rectangular 
coordinate screws. We may write these in the form 
