30 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
which gives 
(H) • • • 
so that the orbit has the approximate period %ir(l + in 0. To agree 
with Einstein’s result this would require a = c 2 /3, and then when r is large 
the quadratic (11) would approach Hdx\ — %c 2 dx\. The Newtonian form of 
Y therefore leads to results which clash with the classical form of the 
relativity theory. 
The usual form of the quadratic is easily obtained by assuming that 
the Newtonian form of V is merely the first term in the expansion of V 
in negative powers of c 2 ; but the method of the next paragraph is more 
satisfactory. 
§ 4. The Newtonian potential is a solution of Laplace’s equation. If 
this equation is regarded as associated with the form 'Edx* — c 2 dx\, Y 
being independent of x if the analogous equation associated with the form 
'Za rs dx r dx s is, as is well known* 
(17) . . . ... . A 2 Y = 0, 
where A 2 Y is the Beltrami parameter 2a rs Y rs 
reciprocal to (cb r8 ), and 
(18) 
8 2 Y _ ^ fr sy_V 
dx r dx s j \ J fdxj 
being the covariant derivative. 
In the case considered in §3, (17) becomes 
(a rs ) being the matrix 
which leads to 
(19) . 
dW 2 dV 1 (d V\ 2 _ 0 
dr 2 + r dr + 2( Y — a)\ dr ) ’ 
Y-a 
Setting k= —ai= -\c 2 , 2/3 = 3 ma 1 , this gives in place of (16') 
( 20 ) 
d 2 u f 5 m 2 \ m 
W + V~£h‘ 2 ) w = h?’ 
which leads to f of Einstein’s value f for the motion of the perihelion of 
Mercury, a result which is slightly closer to the observed value. The 
corresponding form does not, however, lead to the correct deviation of a 
ray of light in a gravitational field. 
There are other methods of deriving an equation analogous to Laplace’s 
* Of. Wriglit, loc. cit p. 53. 
t The Einstein value is obtained by setting 
