102 
Proceedings of Royal Society of Edinburgh. [sess. 
show to what quantities the operator V refers, both the V and the 
quantities having, for this purpose, the same suffix. Thus below, 
•Tv^v^y.yS^i+vi^s^h... 
1 2 1 2 dx dx dz dy 
dz 
One advantage of this notation is, that the V/s, V 2 ’s, &c., as 
well as the oq’s, <r 2 ’s, &c., can be treated as mere vectors.” 
It is this last statement which points to the most powerful argu- 
ment for not restricting the position of the V ’s in a term in any- 
way. Let them have all the freedom of vectors, and they will obey 
all the laws thereof. As soon as we allow this freedom, but not till 
then, can we draw upon the large storehouse of already accumulated 
knowledge of vectors, f 
It will be observed that in the above definitions there is no abso- 
lute need for introducing the symbol A. It is, however, as in many 
cases of duplicate notation, extremely convenient. Its differentia- 
tions refer to all the symbols in the term in which it occurs. Thus, 
for instance, the above equation (1) can be written 
<£A =YD A@, 
which still more clearly shows the connection between the stress 
V l £)( )(£ and the resulting force per unit volume V£) A Qc. 
With these definitions, Professor Tait’s well-known theorems in 
integration can be thrown into a rather more general form. J If <j> be 
any linear quaternion function of a quaternion, then with Professor 
Tait’s ( Quaternions , 3rd ed., § 482) notation for integration — 
* According to Professor Tait this must be written in some such form as the 
following :—YVi(S 7 2 <ri)a - 2 or - Sv 3 Vi. Yv^S^o^ or - Sv 3 v 1 Sv 2 cr 1 . Yp 3 (r 2 . 
Even if the first of these is chosen, it may be pertinently asked why symbols 
which will, in the nature of things, obey all the laws of vectors, should be 
restricted in a purely arbitrary manner in obeying those laws. Why should 
we not be allowed to ring all the possible changes on the form of Vv 1 cr 2 S(r 1 v 2 
as on that of Y a 3 S 7 S, and reap the corresponding advantages ? 
t In the Trans. Roy. Soc. Edin., xxvii. p. 251 (1874), Dr Plan- suggests 
cIt* cIt* 
the notation ^ r=, ^aj + • • • > + ' * • where r is any quaternion. He 
suggests no notation for . . . where 0 is a general linear quaternion 
function. His symbols < and > do not obey the laws of vectors, the first 
only because it is not allowed the freedom of a vector. 
£ The proof is given in the paper already quoted from. 
