in relation to Centripetal Velocity. 57 



From perihelion to the maximum of w the augmentation in- 

 creases from zero to — -=— , and from the maximum to aphelion 



the augmentation diminishes again to zero. Thus the integral 

 for the ascending half is merely a repetition of the integral for 

 the descending half. 



Let c represent the orbital velocity in a circle with radius a ; 

 and let tan^ = v e —\a — i) _ ^^ ^j w==ctSinp ( as snown 



in Appendix) co = dw = c .dtanp; and 



(o/2a—r } 2a — i*\_c.dtanp/2a — r l 2a— r\ 



V\Fi r~~)~ D \r~ l r~~y 



The relation between r Y and r through p, which is common to 



both, enables this expression to be integrated, as shown in the 



Appendix. Twice the integral gives the proportionate decre- 



— made good during one revolution. 



From this we obtain the proportionate decrements of minor axis, 

 exceutricity, and period, — also the change in perihelion distance, 

 which is an increment : — 



T.-J5-' 



[?KR]. 



[?]=[?](?,+■>« £H?](H> 

 SO -ffil-5 



On examining these expressions, it will be found that when e 

 is small with reference to a, as in the planetary orbits, there is 

 little change in axes or period, but the excentricity decrement is 



x- time the decrement of a. 

 2e 



The following is computed for the earth's orbit, taking U= one 

 million, which is tantamount to assuming gravitation to be re- 

 duced one millionth for each mile of centripetal velocity : 



pi = -0000000334; [— ] = -'000001016; 6/= -l s -58 



