1896-97.] Prof. Tait on the Linear and Vector Function . 311 
leaves unaltered ; if but one, we may take a corresponding system 
in the form 
a, ftcosa ± iysin a, 
where t is J - 1. But it is preferable to keep the simpler form 
a, ft y 5 with the understanding that ft and y may be bi- vectors, of 
the form just written. 
3. In terms of the three roots thus designed, we may form, with 
the help of three arbitrary scalars (two of them bi-scalars of the 
form y ± lz, if necessary) three very simple but distinct varieties of 
linear and vector function : — viz. 
(a) Strains leaving three directions, a, ft, y or Y fty, V ya, Va/3 , 
unaltered, so that their reciprocals have the same form. 
Safty <pp = XaSftyp + y/TSyap + 2ySaft>, 
with Safty <£ x p = xYftySap 4- yYyaSft) + zVaftSyp. 
In this case, if x , y, z are the same in each, <f) l is the conjugate 
of (f>. 
(When x = y = z } these strains leave the form and position of a 
body unaltered ; but each linear dimension is increased x fold.) 
(b) Pure strains : — 
<7Tp = xa&ap + y/^S/Ip + zySyp , 
with trr i p = xV ftySftyp + yYyaSyap + zV a ftSa ftp. 
The second of these changes the system a, ft, y, into Yfty, Yya, 
Y aft ; while the first effects the reverse operation. 
(c) Combinations of two or more, from (a) or (b), or from (a) 
and ( b ) : — 
Either form of (a) repeated (with altered constants) simply 
perpetuates the form. In cf>cf) l and <£ x <£ we have new form: , 
which are pure when x:y :z are the same in each of the factors. 
The two forms (b), in succession, give one or other of the forms 
(a) ; and, conversely, either form of (a) may be regarded as the 
resultant of the two forms (b) taken in the proper order. This is 
the main result of my former paper : — for it is obvious that, having 
between them twelve disposable constants, £7 and Zo 1 may be made 
to represent any two pure strains. 
