200 



STELLAR PHOTOGRAPHY. 



vy •>• fj fit 



When 8 = \ n, -j-t = 0, and -^A = sin y cos i ; in this case p = y, a? = —sin y, 



and # = sin y sin & With the system of co-ordinates which has been employed, this 

 represents the circle in which the celestial pole appears to move round the instru- 

 mental pole. 



When 8 < \ y, an extreme value of y will occur for some value of t between 

 and \ 7r. This is apparent from the values already found for which y = 0. The 

 extreme values of this branch of the curve will be numerically greatest when 8 

 as is shown by (4), where the two terms of the value of y have contrary signs 

 for values of t between and a it. Their difference, accordingly, will be largest 

 when the first term has its least and the second its greatest numerical value. This 

 occurs when 8=0, and the value of y is then — sin 2 J y sin 2 t ; at its extreme 

 values, when t = ^Trort = \Tr, y = ± sin 2 J y. The corresponding maximum and 

 minimum in the other branch of the curve have in this instance the same value. The 



general condition for a maximum or minimum of y appears from the value of -^ to 



be cos / (cos t — \ cot \ y tan 8) = \. This is satisfied by t = ± \ it, if 8 = 0, as 

 has just been shown ; also by cos t = 1, if h = ± y, this result, also, has been 

 sidered above. In other cases, the values of cos t required for the maximum 

 minimum of y are found, by the solution of the quadratic equation just given, to be 



J (cot % y tan 8 ± ^8 + cot 2 i y tan 2 8)- 

 To find the greatest and least possible values of y, we have also, from (4), 



■g-ji = sin y cos 8 sin t -f- 2 sin 2 \ y sin 8 sin t cos t, 



which must vanish for the extreme values required, so that cos t = — cot | y cot 8. 

 Equating the two expressions thus found for cos t, we have 



cot J y cot 8 = — J- (cot } y tan 8 ± ^8 + cot 2 -| y tan 2 <$)> 



and, after reduction, tan 2 8 = tan 2 , 2 _ r By supposition, y is positive, and cannot 

 exceed J it ; hence tan 2 J y never exceeds 1, and no extreme value of y can occur 

 unless y = 8 = i it. Accordingly, the numerical value of the maximum and mini- 

 mum of y for a given value of 8 increases from sin 2 J y when S = iy to sin y when 

 8 = I- ir, without reaching an algebraic maximum unless y = J tt, when sin y denotes 

 an extreme, as well as a final value. 



d 



No material modifications of the expressions already given are apparently 



re- 



quired when the star passes between the instrumental and celestial poles. The equa- 

 tion of the curve in two special cases is given below. 



