Astro-Photographic Catalogue Plates. 117 



^8.^-6^ = ^^6-A,6- r/ . 15 X^ + r/ . 15 X, + j>/ F^-p,' Y, + c/ - c,' 



( —Ih • T5 ^2 sec (^2 + T5 ^1' sec rf^ — ^U fc^' sec dj 



If we now write 



^^ = Aj(5 — ((5' — rfj) Vj^=:A-^a — (a' — Oj^) 



1X^ = ^8— [6' — 6.-^) v^ = A.2a—[a'—a.,) 



N={\^ — \'^)—{^2 — <^i) ^o=(\« — ^2'')— («2 — «i) 



the above equations talce the form : 

 l-r,' . 15 X, +p,' Y, + c/ + /'i^o 



, J W — ''2' • 15 X2 +J)2' F2 + C2' + //2 =- o 



) ri' . Xi sec (^1 + j)i' . T5 I'l sec (J^ + 1^5 k^' sec (5^ + v^ = o 

 V r^' . X2 sec (?2 +i>2' • T5 ^2 sec S^ + j's ^^2' sec S.^ -^ v.^ = o 



I -r,' . 15 Xi + r,' . 15 X2 + p,' Y,-p,' Y, + c/- c,' + !x, = o 

 (II) +*'i' • Xisecf^i — r^' . X2secc52 + Ih' ■ iV ^iSeccJi — jja' . Jg FaSecf^a 

 ( -r T3 ^"i' sec ^1 — xV ^'2' sec ^2 + V(, = o 



In these equations p.^, /j.^, v^, v^ can be computed for the known 

 stars, and p.^, v^ for the unknown stars. It is therefore clear that 

 every known star on each plate will give us a pair of equations of 

 the form (I); and every unknown star on the overlapping portion 

 of the two plates will give a pair of equations of the form (II). By 

 combining all these equations by least squares, or otherwise, we 

 can arrive at values of ^/, r/, etc., which, being added to the 

 former provisional values, p, r, etc., give us the definitive con- 

 stants of reduction. 



In the above operation we should of course use all the equa- 

 tions of the form (I) that can be obtained from the known stars. 

 The number of equations of the form (II) which must be in- 

 cluded will depend on the precision of the photographic measures, 

 relative to the precision of the places of the known stars. If the 

 photographic measures are very much more accurate than the 

 places of the known stars derived from meridian observations, 

 then a very small number of equations of the form (II) will 

 suffice. It should also be noticed that experience will perhaps 

 show that p^' and p/ are either equal, or connected by a very 

 simple law depending on the temperature of the telescope tube. 

 If so, our equations will of course be much simplified. In order 

 to settle this point practically, it will be necessary to compare 

 the values of p derived from many different plates. In doing 



Annals N. Y. Acad. Sci., IX, April, 1896.— 9 



