310 Proceedings of Royal Society of Edinburgh. [sess. 
On the Linear and Vector Function. By Prof. Tait. 
(Abstract.) 
(Read March 1, 1897.) 
In a paper read to the Society in May last, I treated specially 
the case in which the Hamiltonian cubic has all its roots real. 
In that paper I employed little beyond the well-known methods 
of Hamilton, but some of the results obtained seemed to indicate 
a novel and useful classification of the various forms of the Linear 
and Vector Function. This is the main object of the present 
communication. 
1. It is known that we may always write 
4>p = ^(aSa^) 
and that three terms of the sum on the right are sufficient, and 
in general more than is required, to express any linear and 
vector function. In fact, all necessary generality is secured by 
fixing, once for all, the values of a, ft y, or of cq, ft, y v 
leaving the others arbitrary : — subject only to the condition that 
neither set is coplanar. Thus as a particular case we may write 
either J 
<f>p — 2a S ip, 
or 
cf >p = 2^’Sajp. 
In either case we secure the nine independent scalar coefficients 
which are required for the expression of the most general homo- 
geneous strain. But forms like these are relics of the early stage 
of quaternion development, and (as Hamilton expressly urged) 
they ought to be dispensed with as soon as possible. 
2. A linear and vector function is completely determined if we 
know its effects on each of any system of three non-coplanar unit- 
vectors, say a, ft y. If its cubic have three real roots, these 
vectors may, if we choose be taken as the directions which it 
