200 STELLAR PHOTOGKAPnT. 



When S = i 77, ^ = 0, and ^ = sin y cos t ; in this case 2^ =^ y, ar = —sin y, 

 and y = sin y sin t. 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 < 2 y, an extreme value of y will occur for some value of t between 

 and I 77. This is apparent from the values already found for which i/ = 0. The 

 extreme values of this branch of the curve will be numerically greatest when 8 = 0, 

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

 for values of t between and J- tt. Their diiference, accordingly, will be largest 

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

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

 values, when t = ^ v or t = ^ n, ?/ = ± sin^ | 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 ^ — i cot J y tan 8) = h This is satisfied by ^ = ± | 7r, if 8 = 0, as 

 has just been shown ; also by cos / = 1, if 8 = J y ; this result, also, has been con- 

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

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



J (cot 1 y tan 8 ± VS -f cot^ i y tan'- 8)- 

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



j| — sin y cos 8 sin ^ 4- 2 sin^ ^ y sin 8 sin t cos i, 



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

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



cot 1 y cot 8 = — ^ (cot I y tan 8 ± Vs + cot- i y tan"'^ 8)> 



and, after reduction, tan^ 8 = ^^^, _ ^ . By supposition, y is positive, and cannot 

 exceed } it ; hence tan^ ^ y never exceeds 1, and no extreme value of // can occur 

 unless y = 8 = I TT. Accordingly, the numerical value of the maximum and mini- 

 mum of 1/ for a given value of 8 increases from sin^ ^ y when 8 = |^ y to sin y when 

 8 = ^ TT, without reaching an algebraic maximum unless y = i tt, when sin y denotes 

 an extreme, as well as a final value. 



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. 



