232 
Proceedings of Roycd Society of Edinburgh. [sess. 
As bearing upon this subject let us inquire bow Professor Gibbs 
would write tbe quaternion equation 
V.qa{^a6)q-^ = i) 
where a may be any constant vector, and where the object is to 
find q and 6. This is one of the latest forms in which Professor 
Tait has expressed the historic problem of finding series of orthogonal 
isothermal surfaces. To represent it in Gibbs’s notation we must 
first take the scalar and vector parts separately, for y has no place 
in his system. Then to express q[ )q~'^ we must take some two 
vectors S, e, in terms of which the dyadics are to be expressed. One 
of these may be any vector in the plane perpendicular to the axis of 
q ; so that in so expressing the versor we are really giving more 
than ought to be required. We then find for the compact and 
most expressive equation just given the following two 
V.{S8-I}.{ee-I}.aa.d = 0 
V X {S8-I}.{ee-I}.aa.0 = 0 
in which V. and V x act on both 6 and e, and where a may be any 
constant vector, say one of a rectangular system. The expanded- 
quaternionic forms 
SV.Se(aSa^)e8 = 0 
W.8e(aSa^)eS = 0 
are even more compact than these. We willingly leave to Professor 
Gibbs himself the task of translating the expression 
where the suffix means that the second V is to act only on the q 
which immediately follows it. 
It has been already noted that Professor Gibbs refuses * to have 
anything to do with tbe quantity Vw where w is any vector. He 
uses the expression in the sense of a dyadic, that is, as an operator. 
Thus 
dp.Vo) 
.do) . jd(ji)\ 
ay dzj 
— doi 
If we except one extraordinary case to be noticed below. 
