346 Proceedings of the Royal Society 
By the value (12) of m 1 the factor of p vanishes. There remains 
(14) - yf> yp = (</) 2 - m.fp + m^)p = $p . 
This gives us also 
£YAp,= — yfyXpi , 
and the second member is precisely the value (11) of Y <f>' \cj>' p , . 
We have thus verified the equation (9) in the case m = 0. In 
like manner we may verify the corresponding equation 
(15) Y = (cf>' 2 - m 2 (f>' + m 1 )YA/x 
derived from : 
(16) -yS y x p = (ft 2 + m 1 )p = £'p. 
These relations (14) and (16) when treated respectively by the 
operator and </>' will give the corresponding cubics (1) in which 
m — 0, because the terms 
4>yfyp, and fiySy^ , 
vanish whatever be p, in virtue of the equations (8) to which </> and 
y>' satisfy. 
These relations (14) and (16) cannot give the inverse functions 
y>~ 1 or because from 
V = ( m 2 - $)p ~ ^"ViSyp 
V = ( m 2 - - </>'“■ V S 7iP 
we cannot conclude anything so long as we are unable to determine 
the unknown vectors 
WXri) anc l <£' -1 (7)> 
in the case when 
^(71) ~ O’ and ^ (7) = 
§ 3. On the other hand, when p is coplanar with the plane to 
which the directions of </>p belong, then calling p 0 such a vector, we 
have : 
S yPo = 0 , 
and consequently 
&>o = W* 2- m 2 ^ + m i)Po = °? 
so that the equation (14) becomes in this case a guadric. 
This remark enables us to construe the equation (14) by a direct 
process. Let namely vector p, of any direction, be decomposed into 
