104 
Proceedings of Royal Society of Edinburgh. [sess. 
vector, whose coordinates are P, Q, R, L, M, H, ~(T is defined as 
meaning the symbolic self-conjugate linear vector function of a 
vector, whose coordinates are — 
d d d d 1 d x d 
dP’ dQ’ dR’ 2 dL ’ 2 dM ’ W 
This twofold meaning for a leads to no confusion, and is convenient, 
as all the formulae which are true of one a are true of the other 
also. To a may he attached a system of suffixes for the same objects 
and with the same meanings, as in the case of V . 
It is convenient, though not absolutely necessary, to introduce one 
other symbol. Suppose Q(a ,/3) is a function of two vectors, linear 
in each. Then w T e have 
Q( V llPl ) = Q(m) + Q (J,j) + Q .(*,*), 
so that V j and p x may be interchanged. This particular use of V 
is so frequent that it is advisable to use a symbol which will call 
attention to the symmetry. I therefore define £ by the equation 
Q( v i>ft) = Q(fi>fi)- 
Similarly, if Q(a,/?,y,8) be a function of four vectors, a,/?,y,S, linear 
in each, we may put 
Q( ^ vPv ^ 2 ^ 2 ) = Q(^1?^15^2^2) ’ 
and so to any number of pairs of £’s. Of course if there be only 
one pair the suffix may be dropped entirely. 
I now state a number of theorems in pure mathematics which will 
be useful, for brevity leaving them unproved. 
If <f> be any linear vector function of a vector, and <f>' be its con- 
jugate, 
-£Sw<££ ........ (4), 
(<j> - $')(!) = VV£<££.to ...... (5). 
If Q(a ,/3) be any quaternion function of two vectors, a and (3 , 
which is linear in each, 
Q(M£)-Q(«,i) ...... (6). 
To apply this in practice, it is convenient to remember it in words : — 
In any term in which £ and <££ occur, ice may , without altering the 
value of that term , substitute for them <£'£ and £ respectively . (It is 
