532 
Proceedings of the Royal Society of Edinburgh. [Sess. 
we get from (9) and (10) 
We shall generally write these equations in the form 
q^rj^rj^Kq . ... 
q = S#Kg 
It may be noticed that if 
a -i — q? a_ 2 = t 2 , .... a_ n = i n 1 
then a_ 1 = t 1} a_ 2 = i 2 , . . . . a_ n = t n 1 
( 24 ) 
( 25 ) 
( 26 ) 
( 27 ) 
( 28 ) 
Putting g — pTS where p is a tictor in (26) and therefore c = n— 1 we 
have, after multiplying into tzr -1 , 
P 1 1 P 1 1 P * * * * 
• (29) 
which is wanted below. A similar more general set of equations is 
obtained by changing q c of (26) to q c T8 and multiplying into Ji5~ x . 
Multiplication of two factors q and K r and the taking of the scalar part 
is so frequent, so simply connected with ordinary algebra [(If), (18), § 2], 
and so important, that I think it desirable to introduce an alternative for 
K, chiefly to be used in this connection. Thus replace K by a vertical 
stroke | , or K = I , 
S.pqKfix) = S£>g|rs (30) 
In this connection I would re-state a plea from quaternionists addressed 
to other users of vector methods. “ Scalar product ” and the symbol S 
should be restricted to one use, and the quaternionists have the priority 
in claim. Why not use the term “ Energic [or else “ ergetic ” or “ ergic ”] 
product,” or the term “ projective product,” for the Cartesian x 1 x 2 -\-y 1 y 2 J rz 1 z 2 
and the corresponding symbol ^p x p 2 ? 
7. Fictorlinities and Multilinities. — [Permittivity, resistivity, and 
their congeners are vectorlinities. The word or termination “linity” is 
understood to imply more than merely “ linear function.” It is understood 
that fictor or vector, or the like, is mentioned twice in the definition “ a 
fictorlinity is a linear fictor function of a fictor ; a fictorcolinity is a self- 
conjugate fictorlinity.” Matrix does not seem to imply so many of the 
properties to be connoted as fictorlinity.] 
In this section I put in systematic form a summary of what I regard 
as the best presentation of the properties of Unities as they appear in our 
present subject. I shall not in proofs go over ground well covered by 
chapter iv. of Octonions, and shall be as brief as possible when transferring 
to present generalised form theorems from section ii. of “ Utility of 
Quaternions in Physics.” 
