160 
Proceedings of Royal Society of Edinburgh . [sess. 
On the Linear and Vector Function. By Prof. Tait. 
(Read May 18 and June 1, 1896.) 
[Abstract.) 
In the following Abstract I refer to such Linear and Vector 
Functions, only, as correspond to homogeneous strains which a 
piece of actual matter can undergo. There is no difficulty : — 
though caution is often called for : — in extending the propositions 
to cases which are not realizable in physics. 
The inquiry arose from a desire to ascertain the exact nature of 
the strain when, though it is not pure, the roots of its cubic are 
all real : — i.e. when three lines of particles, not originally at right 
angles to one another, are left by it unchanged in direction. 
]. The sum, and the product (or the quotient), of two linear 
and vector functions are also linear and vector functions. But, 
while the sum is always self-conj ugate if the separate functions 
are so (or if they be conjugate to one another), the product (or 
quotient) is in general not self-conjugate : — though the determining 
cubic has, in this case, real roots. The proof can be given in many 
simple forms. 
If iff and to represent any two pure strains, there are three real 
values of g, each with its corresponding value of p , such that 
&p=gap, ..... (1) 
Assume io h p = a - ; and the equation becomes 
r = gcr. 
But tulwar* is obviously self-conjugate. Hence the three 
values of g are real, and the vectors tr form a rectangular system. 
Thus (1) is satisfied by three expressions of the form 
p = oThr = g^ffi-^cr ; ..... (2). 
i.e. there is one rectangular set of vectors which have their direc- 
tions altered in the same way by the square roots of the inverses of 
each of the given strains. 
