216 



where p is a very small quantity. Then since the equa 

 tion 



is always true, we have 



ESin. = 



Substitute for B. and Sin a their known values 

 d s r d 



&:*&**$+$ 



where j3 is the angle the tangent makes with any fixed 

 straight line, 



+ constant. 



But /3 = a, hence the variation of a is expressed by 

 the above integral. As an approximation, consider p as 

 a small constant whose square may be rejected, hence 

 a =a +pQ 



Now p is so small that it requires a large value of /3, the 

 angle described, to render the latter term sensible. Hence 

 for many revolutions we may regard a as constant, that 

 is, regard the orbit as an equiangular spiral. We may, 

 therefore, apply our preceding conclusions. If the radius 

 of a planet s orbit be r, by equations (3) (4) (5) we learn 

 that the time before the radius has decreased by S r will 

 be 



T = 1 8 r 



2V Kr g 



and the number of revolutions in that time will be 



AT l * r 



JN = - . = nearly. 



4 TT x r 2 



The value of x is so small that, as these formulae show, 



