Ray of Light in the Solar Gravitational Field. 587 



since values between these roots make the cubic negative), 

 the sign of the radical changing as u passes through the 

 value u 2 . Accordingly as u varies from u to u 2 , the angle cp 

 traces out the angle to the perihelion from a remote distance. 

 The excess of twice this angle over it gives the amount of 

 deflexion of the ray. Solving for dcp, we have as the 

 required deflexion 



D = 2p 2 du 



•-'0 



mir 



§ 3. Evaluation of the Elliptic Integral for the Deflexion. 



It is convenient to make a simple linear change of 

 variable under the sign of integration. We write u = a + bz 

 and determine the coefficients a and b of the transformation 

 so that to the roots u Y and u 2 of the cubic will correspond 

 values and 1 of z respectively. The values are a = u l ; 



b = u 2 — u 1} and then the third root u 3 goes over into y^, 



where P = (u 2 — Wi)/(w 3 — %). The cubic 2mu d — u 2 -\-a 2 



transforms into 2mb d z(l — z)(j^—z\, so that 



dz 



J'-Amh) . 



^2mbJ. a;b v / z(l-z)(l~k 2 z) 



This simplifies considerably on writing z = sin 2 (9, when 

 in fact, 



D = U f ** (W 



Now P = — 1 == — (to a first approximation) 



Uo — Ui 1 



2 m 



is a small quantity of the same order of magnitude as a. 

 Hence we can expand (1 — Psin 2 #)~2 j n a rapidly con- 

 vergent series, and a mere integration of the initial terms 

 will give a very good approximation to D. The multiplier 



of the integral in D is 4,k/\/2mb = 4 a / - — 7 r from 



the expressions for k and b. From the values given for u. 6 

 and u x this is 4(1 + 2ma) - *= 4(1 — ma). The lower limit 



* / ii, 



of the integral is sin -1 \/ — == sin _I \/^(l — ma.), where 



V u x — u 2 



