14 



C. V. L. Charlier 



9. To determine the relations between I, m, n and ^, t], C is hence equivalent 

 to determining the relations between the right angular sphaerical coordinates of a 

 point according to two different systems of coordinates. 



We have generally 



i = I cos X3 -(- w cos YE -(- n cos ZE, 

 (14) r] I cos XH -|- m cos FH -j- w cos ZH, 



C = / cos XÏj -\- m cos FZ -)- w cos Z7j . 



The coefficients of m, w in the right members may be expressed through the 

 ExiLERiAN angles ß, N, N' (compare fig. 3). We get 



(15) 



cos X3 : 



cos YE ■■ 



cos ZE : 



cos XH 

 cos Yti 

 cos ZH 



cos X Z 

 cos FZ 

 cos Z /j 



cos iV cos iV' + sin N sin iV' cos ß 

 sin N cos N' — cos N sin N' cos ß 

 sin iV'sin ß 



cos N sin JV' — sin N cos iV' cos ß 

 sin N sin N' -f cos iV-cos iV' cos ß 

 cos iV'sin ß 



sin N sin ß 

 cos N sin ß 

 cos ß. 



Fig. 3. 



The orientation of the X-axis, from which is reckoned, may be chosen 

 arbitrarily. For simplifying the formulae we put N' = 0, so that the H-axis coin- 

 cides with the descendant node of the EH-plane on the XZ-plane. We further put 



iV+90« = a, 



