NOTE ON Einstein's tlanetary equation. 15^3 



The three special cases in which the eUiptic function 

 degenerates will perhaps give the clearest idea of the shape of 

 these inner orbits. 



I.— If a=iS=y=l'6m (in which case l/h^^Sa^--! /12m2, 

 and (l-c2)/2mh2=a3, :.c-=^) ■ 



.-. u - a = 2/m^-, a .spiral only reaching u = n when f> = co . 



II.— If a=-/3 



/9a/-- i r ^^=- = r — ^^ = -7^ooth- V 



V i: - y 



-r 



-y 



(u_y) = (a-y) coth- (y v''^m(a - y) ), again a spiral leading to 

 u = a when 6^ = ©o . 



III.— If ,/3-^y: 



^V^=i y^ Vt"^a(t-y) y r- + (a-y, v/a - y Va-y 



^U - a 



.•. (u — a) = (a-y) cot'^ (9Vim(a — t)) "^'^ich reaches its apse 

 u = a when 6 = tt/ ^2m(a - y)- 



Of course, these results suggest visions of spiral nebulas, 

 corona, prominences, etc. But the following considerations dis- 

 perse them rapidly. 



§ 4. — Measurements of our solar planetary motions show that 

 m=l'47 kilometres. [It will be noticed that the system of units 

 adopted in the fundamental equations, in which the velocity of 

 light^l, makes m and h of dimensions 1 in length or time, c 

 being a number: and 1 sec.=3.10^km., so that, for instance, the 

 radius of the earth's orbit = 500 sees.] 



m=l'47 km. leaves no room for our "inner orbits" in the case 

 of the sun. But there is no reason to suppose that the same 

 objection would apply to other planetary systems : and there is 

 no intrinsic objection in the theory to m being a minus quantity, 

 and a, /3, y being minus, or P, y conjugate imaginaries. The 

 latter assumption would not seriously affect the above results, 

 though the ordinary outer orbits would be impossible. 



