172 
[Sess. 
Proceedings of the Royal Society of Edinburgh. 
and putting p = ^i gives (because = 9i/^i) 
S//3i = 0, S\'p, = g,-g (11) 
The constant vector jm', being thus at right angles to both /3 and must 
be parallel to Y — 
P=cY/3l3„ ( 12 ) 
where c is a constant scalar (perhaps null). Again, the constant vector X', 
being at right angles to must be of the form c-^Va^, where a is a con- 
stant vector. The relation SX'/3^ ~9i~~9 gives 
c 
^ Sa^/?; 
Substituting in (9) the values of X' and jm', 
4>P = 9P + c/3Spf3^p -H c^/B^SafSp, 
• ( 13 ) 
(14) 
which may be taken as a normal form for a linear vector function having 
hut two distinct axes, one of these being a double axis, corresponding to a 
double root of the cubic in (p. 
It is worthy of note that, so long as /5 and a determine a fixed plane, 
we may alter a in any manner in that plane without altering (p, the 
constant c satisfying the equation 
pa = ga + ..... ( 15 ) 
4. The chief peculiarity of the strain defined by (14) is its effect upon 
vectors in the plane of a and (3. It is evident that any vector in that 
plane undergoes stretching proportional to the factor g, and is further 
altered by addition of a component parallel to /3. The effect of repeated 
operation with </> upon all vectors in this plane is therefore continued 
progress toward the direction /5. No such phenomenon can occur when 
the strain is of the general type. For convenience I shall speak of a plane 
of vectors affected in this manner by a strain ^ as a precessive plane ; and 
(p may be said to be precessive with respect to that plane. 
For a precessive plane to exist, it is necessary that four vectors a, ^ 
(pa, <pp shall be coplanar, while <p has but one (distinct) axis in their plane. 
These same conditions are, in fact, sufficient that <p shall have all 
axes within one plane. For let /3 be the axis in the precessive plane. If 
another axis exists, call it jB^, changed by (p into giiBy We then have (p 
completely determined in the form (14), since we know its effect on three 
diplanar vectors a, /3, and 
Since the most general type of strain depends on nine scalars, and, for 
(14), we have imposed only the limitation that two roots of the ^-cubic 
