167 
of Ediiiburgh, Session 1882-83. 
Looking upon a" - a = d-^a, p" - p = d^j3 as representing any dis- 
placements, we apply the conditions I. and II. of § 1 in order to 
determine the nature of these displacements. Generally then they 
must satisfy the equation 
(46) T(Yq^-p^)TY(d^ad^(3) = 0 . 
Let us examine the second factor. We have as to a, 
d^a = a" — a = (a” — d) + (d - a) = dd + da . 
Thus 
. Y{d^ad^(S) = Y{da + dd)(dj3 -f- d ^') . 
But separately da with dj3 on the one hand, and dd with d(B' on 
the other, give perversions by hypothesis. We have therefore 
T.Ydad/3 = 0, T.Yddd/3' = 0. 
Thus we get 
Y(d^a. d^/3) = V. dd. d/3 + V. dad^'. 
How let us consider the displacements da, d/3, &o., under their 
other form (19), 
da = 2rjSr]a , dd = 2r]'Sr]'d , 
d/3 = 2rjSrj(3, d/3' = 2fSf/3'. 
These give 
Yddd(3 — dYr]'y]^r] d^r]^ , 
Y dad/3' — — 4Y 7]' rjBf /3'Srja . 
Taking the sum, and remarking that d = a + da, /3' = /3 + d^, we 
get for the factor of dYr/yj, 
ST7'[(a + 2ri^r]a)Sr]l3 — {(3 + 2r]Sr]/3)S'qai] , 
= S . r]'aSr]/3 - Brj'ftSrja , 
= S.Yr]yj'Ya^. 
So that 
Yd^ad^/3 = iYrj'7)S.Y7irj'Ya/3 . 
Generally this result cannot vanish, because /3 which is p-~ pc can 
be any vector. 
As this factor of (46) cannot vanish, we must have 
(47) TV(j,-j,,) = 0, 
a solution which gives a screw-rotation, the operators 
(48) ),5,-i and q^{ ) 2 f ‘ . 
