506 Tl. A. Bumstead — Lor entz-Fitz Gerald Hypothesis. 

 The acceleration along the radius vector is 



f n sin <£ -f f t cos 4, = -^4 (1 - PT (I- /5 2 cos 2 ^) 



The acceleration perpendicular to the radius vector is 



V 2 e 1 



f n cos <£ — / t sin <j> = - (1 — ft) 72 sin <£ cos <b . £ 2 



m r r 



If we take the earth as a numerical example, this perpendic- 

 ular acceleration is very small. Its maximum value will occur 

 when the earth is at the extremities of the minor axis of its 

 orbit ; at this point 



cos <f> = e ; sin <^> — ^/ 1 _ £ 2 . 



where e is the eccentricity of the orbit. Taking e = 1*7 X 10 -2 

 and /3 2 = 10- 8 we find 



V 2 e 1 



Acceleration along r = — (l — /2 2 )2 [1 — 2-9 x lO -12 ] 



Y 2 e 1 



Acceleration perpendicular to r = (1 — /f)-2 [1-7 x 10 -10 ] 



7)1 Q V 



I am not sufficiently familiar with the details of astronomical 

 calculations to be able to say with entire confidence whether or 

 not such an acceleration perpendicular to r could be detected. 

 It seems, however, unlikely. The maximum effect is of the 

 same order as would be produced by a perturbing body at a 

 distance equal to that of the sun, and whose mass was only 



that of the earth. The perturbation, moreover, would 



200,000 r ' 



be periodic, vanishing at perihelion and aphelion and acceler- 

 ating the earth's motion in one-half the orbit, retarding it in 

 the other half. 



When the sun is also moving, the problem becomes more 

 complicated. For the present purpose it will be sufficient to 

 obtain the order of magnitude of the acceleration perpendicular 

 to the radius vector. Let v be the velocity of the sun, and u 

 that of the planet relative to the sun. Then the force on the 

 planet is 



(E) = E + (v + u) X H 



where E is given by equation (1), in which (3 is now the ratio 

 of the velocity of the sun to the velocity of light, and 6 is the 

 angle between the radius vector and the sun's path. The mag- 

 nitude of H is given by equation (2). The force E is along 

 the radius vector ; the force (v + u) X H is normal to the 

 resultant path of the planet. Let yjr be the angle between r 

 and the tangent to the resultant path of the planet, then 



