1888-89.] A. MAulay on Differentiation of a Quaternion. 201 
Differentiation of any (Scalar) Power of a Quaternion. 
By Alexander MAulay, Ormond College, Melbourne. 
Communicated by Professor Tait. 
(Read February 18, 1889.) 
Nowhere, I think, does Hamilton, or any other author, attempt 
the very fundamental problem of finding the differential of q n where 
q is a quaternion and n any scalar. It was by noticing an oversight 
in Tait’s Quaternions , § 182, where he considers d . q%, that I was led 
to a consideration of the subject. 
In that section Tait says that the equation 
dq.r -rdq = dr.q- qdr . .... (1), 
where r — ff (2), 
is sufficient to determine dq as a function of dr, hut this will be 
found not to be the case. Equation (1) will be found to be equiva- 
lent to but two equations among scalars, whereas the equation from 
which it is derived, viz., 
q 2 dq + qdq. q + dq.q 2 = dr (3), 
is equivalent to four such equations. Equation (1) may be written 
VVrV dq = VYqY dr , 
and thus it only gives the component of Ydq, viz., V ^r.YYrYdq, 
perpendicular to the axis of r. There are thus two scalars involved 
in dq (viz., S dq and the resolved part of Ydq parallel to the axis 
of r), which, so far as equation (1) is concerned, are left perfectly 
arbitrary. 
In fact, a , b being given quaternions and q a sought one, the 
equation 
aq-qa = b , 
in which the conditions 
S5 = 0, Sab = 0, 
must be satisfied, gives as solution 
q = x + %(y + b)Y- 1 a, 
where x and y are arbitrary scalars. If we use the method and 
result Tait suggests, we are led to dq = oo .* 
[* See Note appended. — P. G. T.] 
