76 



C. V. L. Charlier 



sphere having the spherical coordinates c. and d. The corresponding rectangular 

 coordinates of the projected poiut are 



(43) ^ _ tg S 



cos [ly. — a^j 



Ehminating a, we get 



X' = I 



tg'8 



which equation shows that the projection of a. small circle (parallel to the equator) 

 is a hyperbola. 



For determining, graphically, the position of the apex we have to project on 

 a plane, for each square, the mean motion of the stars. Let Aa and AS be the 

 mean proper motions of the stars (we have put above x^^ = cos SAa, y^^ = AS), we get 

 from (43) 



^x = — J- V 



cos'' a — o...] 

 (44) ^ 



. tg S sin (a — a„) , AS 



cos^ (a • — • a.^) cos^ S cos {'J. — aj' ' - 



Calculating from (43) the projected coordinates of the centre, for each square, 

 we get from (44) the projection of the mean motion of the stars. 



The sphere has in this manner been projected on a parallelopipedon, having 

 a side equal to 2 (the radius being taken as unity) and a height equal to 2tg 66°26'.6, 

 so that the squares J.,, A^, and F.^ are projected on the bottom surfaces. 



On the plates I — V I give the graphical representation, on this parallelopipe- 

 don, of the lines serving to determine the position of the apex, as well as of the 

 both vertices. All calculations necessary to define these lines as well as the graphi- 

 cal constructions themselves have been executed by Mag. W. üyllenbeeg, second 

 assistant of the observatory. 



16. The motion o/ the invariable plane. The equations (41), treated according 

 to the rules of the method of least squares, give the following normal equations 

 for determining F and Ai: 



32.4404 r = — 0". 03245, 



31.7680 M = + 0".iioeo, 



where all stars of the 4"^ and the 5'^ magnitude are taken into account. We 

 thus get 



r = — 0".00100 + 0". 00180, 

 M = -\- 0". 00338 ± 0" 00180. 



The values of the mean errors in F and Aï are here taken equal to the mean 

 error in d-^s from stars of the ö*'^ magnitude. Actually the mean errors in F and 

 A/ ought to be somewhat smaller, as here also stars of the 4*"^ magnitude are ta- 

 ken into consideration. . 



