of Edinhurgli, Session 1882-83. 169 
Multiplying member to member and observing (a')^ = - (Ta')^, 
we get 
, ,,, S'wVSm 
'pp= . 
sin sin 
But the same calcul, proceeding from — will establish 
, 8f/3'Sr]l3 
qq = — — — — • 
^ ^ Sln^; sinu 
Both results give the same operator 
■ m w, 
whatever the signs of the scalar factors may be in the particular 
expressions of pp and q'q . 
Having ^ = rj^=z - , and ^ , we get 
= fr ] . 
We have already defined rjf by (49). From that definition we 
deduce 
(50) 7;'77‘■^ = cos^^; + ^sin^^7 = r, 
so that the operator will have the canonical form. We have thus : 
(51) p"-p"o = >-(p-p>-i 
for all possible values of p. 
The angle lo is evidently by (50) the angle comprised by the two 
planes of symmetry, the angle of rotation being = 2w. 
Let us determine the position of the axis of rotation. By the 
expression (26), in which we change 
p^ into Pc , dp^ into p'\ - Pc = 
= (^a + ^c) + (^a' + ^c') ,, 
and u into w, p into r = 7} , we get : 
^.-1 
Now we have 
Hence 
po-pc^C^ + lV ^-^{rja + fE) + {ia + f a')~| . 
Lsin(^? J 
