242 



7 = 00, as in the approximate solution (16), in which /> was neglected. 

 For the planetary motion we must go to the second order. 1 

 find a motion of the perihelion amounting to 



ffco — — I / a" nf (19) 



which is of course entirely negligible on account of the smallness 

 of A a'. In my last paper ^) it was stated that there is no motion 

 of the perihelion. In that paper the values T „,," were used, i.e. the 

 pressure /> was neglected. The motion (19) can thus be said to be 

 produced by the pressure of the world-matter on the planet. It will 

 disappear if we suppose that in the immediate neighbourhood of the 

 sun the world-matter is absent. 



12. In the system B outside the sun we have (> =r 0, and the 

 equations are dependent on each other and can be integrated. 



Within the sun n'a^i'^ must be of the second order, and conse- 

 quently n' must be of the tirst order. If we put 



ƒ = cos' X (1 + y), 



2 y' tafi X 



then 71 =■ tanx-}-- , thus must be of the first order. 



R 1 +y R 



Since x = ''/^ ^^ ft"d that 1/R' must be of the first order, as in 



system A. 



Developing ƒ in powers of 1/R we find, to the first order 



In the first approximation we find for y the same value as in 

 the systems A and C, viz : y = — a/?'. Here however we have also 

 the term — 'Vii*- Thus classical mechanics according to Newton's 

 law can only be used as a first approximation if this term, and 

 consequently also A == V/?^ i^ of the second order. Investigating the 

 effect of this term on planetary motion, we find a motion of the peri- 

 helion ') amounting to 



3a' 



öio =z nt . 



2&R' 



1) These Proceedings, Vol. XIX, page 1224. 



2) In my last paper (these Proceedings Vol. XIX, p. 1224) I found 



.3a' cnt^ 



öió = nt — . 



4a R' 2R^ 



The difference is due to the use of a different system of reference, with a 

 different time and different radius-vector, in the two cases, the foimulas for the 

 transformation of the space-variables (especially the radius-vector) from one system 

 to the other depending on the time. 



