312 
Proceedings of the Royal 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 
q + mq = a , 
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 d md K 
2. But if a be a quaternion , the integration of 
q + aq = a! 
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 
r = ra , 
or, if r' be a proximate value of r, 
r' = r + rSt — r (1 4 - aSt) . 
Hence, dividing the finite interval t into a great number of equal 
parts, and taking the limit 
where r 0 is an arbitrary but constant quaternion. 
Now we have 
at t(Sa -f- TVa . TJVa) t(m -f- na) 
£ = s ~ ’ suppose 
mt ■7T 
a 
Hence the solution of the given equation is 
