102 Reduction of Stellar PhotograjJhs. 



Some of the writers on this problem have allowed the quantity 

 d' to appear in their formulae : this may be necessary for plates 

 taken very near the pole, but for all other plates it is not necessary. 



To secure the maximum of facility in computation, we require 

 the expansion of Aa and Ad in ascending powers of x and y, in a 

 series whose coefficients involve S only. We also require x and 

 y expanded in ascending powers of Ace and A^, with the same con- 

 dition as to the coefficients. These considerations lead to the 

 following expansions, in which the unit for x* and Aa is the sec- 

 ond of time, and for y and Ad the second of arc. The same rule 

 with regard to units applies to all the other formulae in the pres- 

 ent paper ; so that wherever x or Aa appear, they are supposed 

 to be expressed in seconds of time, and wherever y or Ad appear, 

 they are in seconds of arc. Similar expansions carried as far as 

 terms of the third order have been given by Ball and Rambaut 

 (Trans. Koy. Irish Acad., V^ol. XXX., part lY.) Those given 

 in the present paper were deduced by me from Turner's rigorous 

 formulae (Observatory, XYI., p. 374) and afterwards carefully 

 checked by Mr. Finlay, who very kindly extended Ball and Ram- 

 baut's work as far as terms of the fifth order for that purpose. 

 They hold good up to within 15° of the pole. 



Aa:=x sec (^ +^1 (x sec (^)y ^1 = tan S sin i" 



-{-A.2{x sec 6)y'^ ^2 = tan rfsin^ i" 



+J3 {x sec f5)3 J3 r= — J ( 15 )2 sin2 i" 



+-^4 {x sec 6yy ^4, = — tanrf (i5)^sin''i" 



-\-4iix sec 6)y^ J5 = tan^(5sin3 1" 



-\-A^{x sec ^^y^ Jg^ — 2 tan^(5(i5)2 sin* r" 



+J. (a; sec (5)5 ^, = 1 (15)" sin* i" 



-]-A^{x sec 3)y^ ^8 = tan* d sin* i" 



A(5 — y-|-Z),(.Tsecrf)2 A = — 5-sin2'5(i5)2sin i" 



-^R,(xsec('i)^y A — — J (i5)^sin-^ i" 



+i>3?/3 Z)3 = — Isin^i" 



+Z),(a;secf5)V A^^ — Jsin^ Jtan'J (i5)2sin3 i'' 



+i)5(.Tsecf5)* A = ^3 sin<icos3rf-l-sin''c5cos(?) (i5)*sin'' i". 



-^D.ixsecdyy D6rr=|(i5)*sin*i" 



+I).{xsecSyY I)T = }{i—tan^6) (i5)2sin*i" 



+-D8f Ds = isin*i" 



■*The use of the time-unit for the linear quantity x is to be understood as 

 meaning that the unit for x is the distance corresponding to a second of time 

 at the centre of the plate. This applies also to the unit for y, mutatis mutandis. 



