STARS IN COMA BERENICES. 443 



is not the case with the measured X and Y. The formulae, ar- 

 ranged for calculation to terms of the third order, are as follows : 



(i8) 



Aa — A' sec Jq = -j- AaAJ • tan (5g sin \" ■ 



— A«3 . I./ ( I _ 3 sin2 £?(,") sin2 l" ; 

 AS — F = — Aa2 ■ X sin 2fJ(, sin \" 



— Aa^AcJ • Yz cos 2^0 sin2 i^-' - (19)1 



— A(53 . ^^ sin2 \" . 



The simplicity and elegance of the above expressions are at 

 once evident when we remember that 0^ is the declination of 

 the center of the plate, and is therefore constant for any group, 

 or, in fact, for an entire zone. It is, however, necessary, that 

 the position of the center should be known. As has previously 

 been mentioned, Rutherfurd was careful to have this point 

 coincide with some bright star ; in the case of the Coma Plates 

 the star selected was \2 e (my no. 14). Taking thus the values 

 of J« and Ad from the Catalog der Astronomisclien Gesellscliaft (cf. 

 the " List of Catalogues" in Part I of the present paper), and 

 applying formulae (18) and (19) to each star, the quantities 

 {Aa — A") sec o^and Jo — Fare obtained. I have collected them 

 in Table VI. Since the rectangular coordinates, x and y, were 

 measured from the same star as origin, it is evident that the 

 table will give at once the corrections which have to be added to 

 A'' sec Og and Y, i. e., to the measured coordinates multiplied by 

 the scale-value for the center of the plate, in order to change them 

 to Ja and Jd. It is also plain that the table will be constant 

 for all the plates, and that the corrections may therefore be 

 applied equally well to the mean of all the determinations, as to 

 each one separately. This I have done. 



INOTE. — Equ.'s (12), (17), (18) and (19) were first deduced by this method 

 by Professor Jacoby. See his review of " Donner, Determination des Constants, 

 etc.," in the Vierteljahrsschrift for 1895, P- ^^4' where these series, to terms of the 

 fifth order are given, but without demonstration. Previously, Ball and Rambaut in 

 Trans. Roy. Irish Acad., XXX, P't. IV, had deduced the first two of the above ex- 

 pansions to terms of the third order, but they were obtained by a process entirely 

 different from that shown here. 



(103) 



