1888-89.] A. M c Aulay on Differentiation of a Quaternion . 205 
Lastly, suppose n is negative, and = - m. Thus 
d.q n — - q n d.q m .q n 
= - (q, — n)q n dqq n 
= (q, n)dq [equation (17)] , 
which proves the proposition for all cases. 
The various forms given above for (q, n) thus give so many forms 
for d.q n . Notice the meaning of equation (14). This gives 
Stfl.q^^nSqn-'dq (20), 
which gives the ordinary form for the differential of a power of a 
scalar. If we put q = a vector p, a = dp , we have 
a = pSp ~ J dp a" = pVp~ 1 dp, 
and equation (8) gives 
d.p n — np n Sp- hip + Yp n . Yp ~ x dp . . . (21), 
and this, be it remembered, is true not merely for integral values 
Of 7L 
Not© on Mr M‘Aulay’s Paper. By Professor Tait. 
(Read February 18, 1889.) 
There is, undoubtedly, an omission in § 182 of my Quaternions 
(2nd ed.), but it is by no means so serious as Mr M‘Aulay asserts. 
In fact the solution there given is merely an unfinished one, not in 
any sense erroneous. I sketch briefly the completion of it, as pre- 
pared for the new edition of my book, which is now being printed. 
The equation 
q n ~ 1 dq + q n ~ 2 dqq+ . . , . -bdqq^- 1 = 4>(dq) = dr . . (1) 
gives, as in my book, 
q n dq - dqq n = qdr - drq , 
but this does not make dq infinite. In fact it gives 
2V.Vq h Vdq = 2V.VqVdr ..... (2). 
Now it is easy to see that 
Vq n = QnVq, 
where 
Q» = »( S 2 )»-i - ”^ - ] -” ^ 2 (Sg) B - 8 (TV qf + &e. 
