k r? +. 7? op? h 
— Sted ey ae 
or 20” c EE, 
and from (18) we find, by putting v=—1 and w=c?, 
r? 7 Ane ana ke oe et (LOO 
The formulae (17a) and (18a) lead to the ordinary planetary 
motion as described by Keruer’s laws. We now sball go a step further 
with the approximation. Equation (17a) shows that %/cr? and 
7? + 7? g?/c? are of the same order of magnitude; both quantities 
are small, as the second represents the square of the ratio of the 
planetary velocity to the velocity of light. We shall call these 
quantities (also 1—h/c) of the first order of magnitude, and we 
wish to retain in (17) also the quantities of the second order of 
magnitude. For this purpose we still need not go further in wand v 
7, and are 
of the second order of magnitude, but would give terms of the third 
order of magnitude in (17), because they occur there multiplied by 
than to terms without x, as £ and y contain the factor x 
r? and 242. The motion of the material point will, accordingly, 
not depend on the special properties of the substance of the 
attracting body. 
Let us now put for brevity 
h 
1——-=l/, w= c*(1--d+8), 
C 
in which / and d are of the first order, ¢ of the second order in z. 
We now expand the root in (17), and omit terms of higher order 
than the second; this implies that in the terms of the second order we 
may apply equation (172), ie.: 
r? ay 2 
See ey ae 
C 
in order to eliminate 7” + rg? from the terms of the second order. 
The result is 
rit pg? = — 2c (1+3l) + ed(1 44) — et(etd"). . (17%) 
To proceed a step further with the approximation in (18), we 
need only put v=—1 and w= c?(1—4); this gives 
pip = Ace a OL Nee ay 
In connection with this we may write for (17°) 
(5) + aoe ee 
r\dp) | r At At? | Ate? 
As ws=1 we get 
1 
heit {1+d-+(d?—e)}. 
67 
Proceedings Royal Acad. Amsterdam. Vol. XVII. 
