687 



[XX X 



A ?i + — sin 6 + 0.38 — cos^' (f sin d + 0.13 — sin" ó -f 



Y Z 



+ 0.05 -- cos-' ösin ö + 0.52 — cos 

 R,. R^ 



'' 6 sin cf co« « 



X Y Z 



04 — cos Ö sin'' d + 0.26 — cos dsirr <f -f 0.10 -- cos' d 

 R„ R,. -R„ 



X Y Z . 



0.26 -— cos ösin" d — 0.04 — cos ösin' d + 0.24 -— cos' 

 R„ R. R„ 



d cos 2« 



+ 



0.05 -- cos'' (f sin d + 0.12 — cos'' d sin d 

 A'„ ^ -R^ 



12 — cos'' dsind 

 R„ 



0.05 — cos^ ösin d 

 ■ R. 



3« 



3« 



In many cases it is convenient to iiiodif j the formulae so that in 

 place of R„ they contain (lie mean distance R,„ corresponding to 

 the magnitude or tlie mean magnitude under consideration. We will 

 define this mean distance as tiie reciprocal value of the mean paralla.x, 

 and therefore put : 



R. 



R„ 



1 +0.65 X mean value «m 'jj 



We must tiien integrate sin^S over the whole surface of the sphere, 

 and in this waj we find: mean value ot' sin^^ ^= -^, so that Ro = 

 1.22 R,n, and this relation must be substituted in all the terms which 

 are dependent upon the parallactic motion. 



To save space, we give below only the values of the numerical 

 coeflicients in the new formulae containing /?,,,. 



Coe/Jicients in the formulae conkdiihig R,,,- 



(Ic COS d =z: 



+ 1.00 — 0.02 + 0.11 



+ [+ 1.00 + 0.82 + 0.11 + 0.11 — 0.04] d7i a 



— [_)_ 0.82 — 0.04 + 0.31 + 0.11 ] cos a 



— [+ 0.11 — 0.02 ] ^in 2« 

 + [+0.02 +0.11 ] cos 2a 

 + [+0.10 —0.04 ] sin 3a 



— [+0.04 +0.10 1 cos 3a 



