168 
Proceedings of the Royal Society 
becoming identical, because the ambiguity disappears in 
the expression of the operator. 
Having arrived at the condition ( 47 ), by way of exclusion we 
will now prove by a direct method the identity of the two 
operators ( 48 ). 
In the preceding deductions we have admitted that p and q have 
the same axis, situated in a plane perpendicular to t;, this plane 
having the definite position of the plane of symmetry. Likewise, 
we have also admitted that the axis of p' and f be situated in the 
plane of symmetry corresponding to the second perversion. As 
these two planes generally intersect each other, we take the direction 
of the line of intersection for the axis ^ of the four versors y>, 
p, q. We have then 
sc>? = o, s^v = o, 
and in putting 
(49) 7/77' =- cos ?<; + ^sinw, 
we assume that the positive half of ^ will be directed so as to have 
77' on the left of 77, when the angle w be positive and not outpassing 
two right angles. Having 
a =pap'^^ a'p=pay 
whence we deduce S^a = S^a, and putting 
a = V(^VaO, cT=V(CVa'0, 
where d and a represent the projections of a and a on a plane 
perpendicular to we get 
This gives Ta = Td. Hence by ( 42 ) we have 
a!p = ^ 
^rjoYa 
sill u 
Likewise we get, applying ( 42 ), 
p a 
— 877'^ Ta' 
Sin u 
