General Relativity and Einstein's Theory. 411 



The corresponding- Lagrangian equations are 

 d /9H\ _dH _ 

 Jl \9f) '9r ~ ^ ' 

 d^/9H\_9ff_ 



d^l\9d) 9e~^' 

 d /9H\ 9fl _ 



Evidently every term in the last equation will contain either 

 <f) or '(f>. Hence if the coordinate system is so oriented that ^ is 

 initially zero, 4> will be zero, and therefore the motion will be 

 confined to the r 6 plane. Forming the derivatives involved in 

 the first two equations, and solving for r and 6, 



k' h'' 



'+~:'=-H^-'B 



Substituting the values of h and k contained in (62), and 

 changing the independent variable from / to ^ by means of the 



relation 



I ^ i c t, 

 it is found that 



;-,.^.= -^(.--^';;-) + ;':(^;--,'»=), (64) 



e+rr6^4^^rd. (65) 



Multiplying (64) by r and (65) by r^6, adding, and integrating, 



+ <'■-' + '■' ^') = '-' (' - - ^ "■''' (,' - ;^) + ^'"f^^ *•' 



if the velocity of the particle is zero when r equals a. Taking 

 the first term on the right as an approximation to value of the 

 left hand member, substituting in the integrand of the last term 

 on the right, and integrating, 



i (;■+ .■ h = «.' (J - ^) - »' r (/, -1 + ^) (66) 



Comparing this integral of energy with that obtained on the 

 Newtonian theory, it is seen that mc^ is the mass of the attracting 

 body (the sun in the case of the solar system) in astronomical 

 units, i. e. the unit of mass being taken as that mass which exerts 



