428 KRETZ. 



6 the angle XOX' , positive if the plate must be turned counter- 

 clockwise in order to make OX and OA^' coincide. Then the 

 positive OX' will fall between positive OX and OV, for, on 

 the plate, positive coordinates correspond to the usual position 

 of the axes, i. e., positive x to the right, and positive/ up, when 

 the trail is left (corresponding to x reversed). 



Let also x^ , y^' \ x^, y^ be the coordinates of the central star 

 referred to the center of rotation, o, corresponding to the actual 

 and to the corrected position of the plate respectively. 



Then by the usual formulae for the transformation of coordi- 

 nates, we have 



Xq-\- X =z (xq'' -\- x') cos — (_)'g'' -\-y^) sin 

 y, +y=W + -^0 sin e -f 0/ +y') cos d 



or expanding and remembering that 



Xq = Xf/ COS 6 — jj/g'' sin 6 

 y^ = Xq'' sin 6 -\-yQ'' cos 



and that 6 is very small, we find 



X = jf' — J)''. 0^^ sin i^'' 

 r ^y' -I- x'. Qf' sin \" 



i. e., from the measured x' we must subtract j'. ^'' sin i" and to 

 the measured j' we must add x' . d" sin i" in order to obtain the 

 correct coordinates.^ It will be seen that these formulae take 

 account of the fact that the center of rotation of the plate does 

 not coincide in position with the origin of coordinates. 



To determine what sign to give 6" in any special case, we 

 need but remember that in the Repsold measuring machine 

 an increase in angle corresponds to positive (counter-clockwise) 

 rotation of the plate. Hence if we let 



Q = the seconds of the circle reading to which all the posi- 

 tions are to be reduced ; 



* See in this connection Harold Jacoby, " Permanence of the Rutherfurd Photo- 

 graphic Plates," Annals New York Academy of Sciences, Vol. IX, p. 267, 

 where the same formulae are given. 



(88) 



