142 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



envelope, Fig. 25. The hydrodynamical illustration of this problem is 

 a lawn sprinkler or fountain from a ball pierced with holes. 



Let us now treat the motion of a planet about the sun, using 

 the coordinates r ; #, cp, defined as in 21. Since r is not constant, 

 we have to use the element of arc 



132) ds 2 = dr 2 + r 2 d& 2 + r 2 sw 2 ftdcp 2 . 

 From which 



133) T = f [r ' 2 + r 2 & 2 + r 2 sin 2 # - cp' 2 }, 



als 



by 28 (y being taken positive). Let us for simplicity take the 

 mass of the planet as unity and write yM=k 2 , we have then 



dT dA 



-t n A\ 



134) .p 9 =, w 



8T , cA 



Accordingly our differential equation becomes 

 JT 1 



= 



Let us undertake to find a solution in the form 

 136) A = R(r) + F(,q>), 



where the functions R and F contain only the variables indicated. 



= = 



dr~~dr' d ~ d&' 



Substituting in equation 135) we have 



1q7 v 1 



Multiplying by r 2 and transposing 



1QQ\ X 



138 ) 



On one side of this equation we have functions of r alone, on the 

 other functions of # and <p alone. Since r, # and <p are independent 

 variables this cannot hold identically unless each side reduces to a 

 constant. The partial differential equation thus falls apart into the two 



139) i. 



