313 
of Edinburgh, Session 1870 - 71 . 
the arbitrary quaternion constant r 0 having disappeared, but a new 
one being introduced by the integration on the right. 
When a is variable, the tensor of r is easily seen to he % fSad* } 
but its versor, s, is to be found from the equation 
s = sY a 
the fundamental relation between the instantaneous axis and the 
versor of rotation of a rigid body (Trans. R.S.E., 1868). 
When r is a vector, 0 suppose, we have 
6 = Y 6a , 
whence, as above, 
e = y e 0 sf adt . 
3. In the succeeding examples we restrict ourselves to equations 
for the determination of unknown vectors , as we thus avoid the in- 
troduction of the quartic equation which has been shown by 
Hamilton to be satisfied by a linear function of a quaternion , 
This would appear, for instance, in the solution of even the simple 
equation 
q + aqb = c 
where a and b are constant quaternions ; though, of course, its use 
may be avoided by employing a somewhat more cumbrous pro- 
cess. 
4. Suppose we have 
p + <Pp ~ a 
where <p is a self-conjugate linear and vector function with con- 
stant constituents. Operate by S . S, and we have 
SSp + S . p(pS = SSa . 
The left hand side is a complete differential if 
S = <pS . 
The general integral of this equation may be written as 
s= 
where s $ is another linear and vector function ; but it is not neces- 
sary to discuss here the validity of such a result, deduced as it 
must be by a process of separation of symbols. [See Tait’s Quater- 
nions, § 290.] For, on account of the properties of p, we may 
