STARS IN COMA BERENICES. 441 



Substituting from (lo) in (9) we get after slight reductions 



^= tan (a — Cq) cos/g [tan/^ — y] , 



or since 



cos /q = sin Jq, sin p^ = cos rfg, . 



a — «Q = Aa, 



-^ JT sec cJ„ 



tanAa = r ~—. — r-=: ~ — ^- - (li) 



cos Oq — v sin Oq I — r tan Og ' 



Apply to this last expression the formula 



tan~l Z( :::=U — ■ ^ u'^ --{- i tt^ . . 



and expand each term by division. To terms of the fourth 

 order the resulting series will be 



Aa = Xsec i5q-\- A^[X sec (')q)Y ^j=:tanrfQ, 



+ A,{X sec '^,) Y^ A, = i2.nk\, 



+ A^{Xsec6,Y A^= — \, I (12)1 



+ A^{X sec (\Y Y A^= — tan Jq, 



+ .'^5(Xsecf5(,)F3 ^5=tan3rfo. 



The process may easily be continued to any number of terms ; 

 but for most cases even terms of the third order are almost in- 

 appreciable, and no accuracy is added by carrying the compu- 

 tations further. Higher terms will be necessary only when 0^ 

 becomes large, or when the plate covers more than 2° square. 



Let us now seek to find a similar series for Jo. The method 

 is entirely analogous to the preceding, but the algebraic work 

 is much more intricate. For we cannot now eliminate and 

 thus get rid of that quantity once for all. We must keep it in 

 the reductions until the end, and then eliminate it by the relation 



Let us consider again the expression for X in the form (9). 

 Remembering the last of equations (8), we can transform this 

 as follows : From (8) and (9) 



j^^ sin(a — gp) tan/ 



cos /g -(- sin pQ tan p cos (a — Qq ) ^ 



From (8) and (10) 



tan />n — Y , , 



tan/cos(a-ag) = ^,^-0^-^. (14) 



1 See footnote, p. 443. 



Annals N. Y. Acad. Sci., XII, April 4, 1900. — 28. 



(101) 



