6o6 PHYSICS: A. G. WEBSTER Proc. N. A. S. 
_A, + _,.^ + ,.y = 0, (3) 
where stands for the Laplacian and for the gradient of q. Dividing 
by dV/dq we have 
Since V and its derivatives depend on q alone, we may write tliis 
Ag + h\-<p{q) + j^) = 0 (5) 
where 
/(?) = log V = //(<7)d<? + C, V = Ae^ f'^^'^ 
d7 = Ae-^''''"M. log ^-^ = log A + y/(g)<ig + log /(g) 
* ) 
Thus we may write (5). 
Ag + .^{/(g)+/^>} +1-^ = 0. (6) 
which is the required condition. This may be put in another form as 
follows: Multiplying by f{q)/h^q we have 
Now although it is not necessary for either Aq or h^q to be functions of q 
alone, since the first two terms of (7) are such, the third term must be, say 
^-^^ = OMfiq)- (8) 
The equation 
^ + + 0/ = 0, (9) 
dq 
is not linear, but may be made so by substituting for / its reciprocal R. 
We then have 
dR 
OR = 1 (10) 
dq 
of which the solution is 
R=j= e^^'^B + fe-^'^'^dq). (11) 
From (11) and (8) we accordingly obtain 
Ag = Qh\ - k'R = Qh\ - h\^^\B + fe-^^''dq), (12) 
