536 
Proceedings of the Eoyal Society 
we have the condition (17) equivalent with SA:'^ = 0. We have 
thus four equations between the three parameters of p( )q by 
which to eliminate (theoretically at least) these parameters, and the 
result will he an equation in p representing a definite surface. 
Practically the elimination may be effected by eliminating h 0 , g 0 
[h 0 , g 0 being the values of h , g in the relations (24) for u = 0] from 
(25), (26), and the third scalar relation spoken of derived from (15). 
We reserve this question for another occasion. 
§ 5 . 
Let us now transform the expression (15) of k' so as to render it 
dependent on h and p alone. 
Taking the function of both members we get 
(29) 0=-^ + (m 2 + u)^>'p - , 
u 
because <£'. Vcfii'fy' — 0 , by considering (23). 
By (10) and (11) we have 
(<£ — <j>')4>'p ~^Vy4>'p = V. (p'k'kfi'p . 
And considering that 'p = &p<f>k' = 0 , developing V.fiic'k'fip , 
and replacing </>'&' by p - uh ' , we get 
g>' 2 p = p + k'$ p4>'p • 
We get SpcJ)‘ p by treating (15) by S. p. This gives, by S kp = - u , 
Spfip — m u + S . pcfii'(f>f + (m 2 + u)p 2 . 
In (29) also we replace <£'&' by p - uk ' and eliminate <j>p between 
(29), so modified, and (15). This gives. 
0 = 
u 
(p - uk') 
+ 
+ u) £ - ' > -~k' + Yc pifif + (m 2 + u)p J 
- 4>(&p) - k'\m u + S p4>i'<hj' + ( m 2 + w )/° 2 ] • 
With the notations of g , h , n (24), (27), this becomes 
(30) 0 = nk' + -zrp + (h - M 2 )p - Y pi'pj'g , 
g being related to li by (25) and n given by (27). 
