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 be ef 8adt J 
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 
0 = V 6 a , 
whence, as above, 
6 = Y6 Q ef 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 . 8, and we have 
S 8p + S . ptpS = SSa . 
The left band side is a complete differential if 
8 = p8 . 
The general integral of this equation may be written as 
a = « ‘H 0 
where 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 Tail’s Quater¬ 
nions, § 290.] For, on account of the properties of <p. we may 
