312 
Proceedings of the Eoyal Society 
vector equations of an order no higher than the second, or to de- 
velope the subject of the curious functional equations which are 
incidentally involved. 
1. The integration of an equation such as 
where m is a scalar (usually a function of t, which is assumed 
throughout as the independent variable), and q an unknown qua- 
ternion, is obviously to be effected by the ordinary method, multi- 
plication by e fmdt • 
2. But if a be a quaternion , the integration of 
even when a is constant, requires a little care, unless we boldly 
treat a as m was treated in the preceding section. This, no doubt, 
gives the correct result, but the process requires to he defended. 
Assume therefore r to be a factor which makes the left hand mem- 
ber integrable. Then we must have 
or, if r' be a proximate value of r, 
r' = r + rSt = r (1 + aSt) . 
Hence, dividing the finite interval t into a great number of equal 
parts, and taking the limit 
q -f mq = a , 
q -f aq = a ' , 
r = ra , 
= U 
where r 0 is an arbitrary but constant quaternion. 
Now we have 
at t(Sa + TVa . TJVa) t{m + na) 
£ = £ = > sum 
’ suppose 
2 nt 
mt 7T 
s a 
Hence the solution of the given equation is 
