Group of the Pleiades. 255 



If we substitute the values from the last equations in the preceding 

 ones, and then in (b), we obtain : 



or — s = -f- k sec 2 z cos 2 l.s — ^ k sec 2 z cot z cos Z sin 2 Z.s 2 



-j- & sin 2 Z.s -j" 2 ^ *an 2(1 + 2 c °t 2 z) cos Z sin 2 Z.s 2 



dk 



■■ -j tan z cos 2 Z.s — & tan z sec 2 z cos 8 l.s 2 



dz 



-\- \k cot z cos I sin 2 Z.s 2 

 — k sec 2 z cot z cos I sin 2 Z.s 2 



s(/t — 1) = — k sec 2 z sin Zcos Z.s -|- ^-&sec 2 zcot z sin 3 Z.s 2 



-f- & sin Z cos Z.s -f -J- & tan z(i -j- 2 cot 2 z) sin Zcos 2 Z.s 2 



cZ& 



tan zsin Zcos Z.s -\- ksec 2 z tanz sin Zcos 2 Z.s 2 



dz 



— & sec 2 z cot z sin Zcos 2 l.s 2 



— ^kcot z sin 3 Z.s 2 



Introducing auxiliary quantities and simplifying, these equations 



can be written : 



— s = s k [tan 2 z cos 2 Z -}- 1 ] -j- A s -j- B s 2 > 



^— l = — k tan 2 zsin Z cos Z -f- ^4' -f £' s J ^ 



where 



^ ^ + 2 7 



J. = — tan z cos 2 Z 



dz 

 B = — & tan z cos Z — & tan 3 z cos 3 Z 



A' = tan z sin Z cos Z 



6?Z 



J9' = 1 & tan z sin Z -f- & tan 3 z sin Z — & tan 3 z sin 3 Z 



The auxiliary quantities A, B, A/ and Z? r , which are very minute, 

 can be taken from the following tables, in using which s is to be 

 considered expressed in seconds of arc. It has not appeared neces- 

 sary to extend these tables beyond the zenith distance 63 , the 

 logarithmic tangent of which is 0.3, since this will probably include 

 all the Rutherfurd photographs of star clusters, as well as any. 

 others that may be made for comparison with them. 



I may add that these tables afford a much more convenient solu- 

 tion than the method previously published by me,* in which the 

 problem was treated in an entirely different way. 



* Astronomical Journal, vol. x, p. 163. 

 ArtNAxs N. Y. Acad. Sci., VI, Feb. 1892.— 18 



