STELLAR PHOTOGRAPHY. 



199 



this may be reduced to the form 



^ = sin 8 [1 — tan 8 cot (8 +/>)] [(1 — tan -X y cot 8 cos <) cot * -f- tan |- y cot 8 sin (J. 



In this expression sin 8 [1 — tan 8 cot (8 -+-/>)] cannot become negative, since 

 3 and /; are positive, while 8 does not exceed \ ir, and when cot (8 + ;>) is positive, 

 tan 8 cot (8 -\- p) is a proper fraction. It reaches the value 1 only in the case of the 

 passage of the star through the instrumental pole. 



Equation (4) shows thaty reaches the value when 



sin \ y sin t (sin 8 cos \ y — cos 8 sin J y cos /) = 



? 



that is, when t = 0, when / = 7r, and when cos t = tan 8 cot l- y ; the last condi- 

 tion requires that 8 shall not exceed -J y, and in that case will occur at some value 

 of t from to \ ir, and at the corresponding value between § 7r and 2 tt. The 

 points of the curve where these values occur are accordingly situated upon the 

 axis, and the general statement of the variations of x, already given, shows that x 

 has the same value at each point. Hence the two points are identical. It ha 



been shown that, in the expression for •**, sin 8 [1 — t;in 8 cot (8 + />)] is positive. 



The remaining factor, since tan 8 cot \ y = cos /, is reduced to 2 sin / cos /, which 

 is positive for the smaller, and negative for the larger value of t. As dx is also posi- 

 tive for the smaller, and negative for the larger value, dp is positive in both cases. 



When t~0, and § < % y, ~j^ = — °° 5 s0 a ^ so when / = v, \i\ — — °°- 



Hence, when 8 < Ay, the curve consists of two closed branches with a common 

 point, resembling a lemniscata ; it will be shown below that, when 8 = 0, the equa- 

 tion of the curve represents a species of lemniscata. 



When 8 = J y, cos t = tan 8 cot a y only when t = ; the lower branch of the 



dingly, disappears. The term (1 — tan A y cot 8 cos t) cot i becomes 



2 sin 2 \ t cot t, which may be written in the form sin £ t (cos \t — sin \ i tan \ t) ; 



if // 



when t = 0, this vanishes, and also the term tan \ y cot 8 sin t ; hence (l - = 0. As 

 this value occurs at the minimum of x, the lower branch of the curve vanishes in 

 a cusp. The value of p in this case is J it + J y. 



When 8 > |y, ^ = only when / = 0or when t — ir. When / = 0, ^ - = oo ; 



when t = tt. 5^ = — oo, as before. For values of ^ near 0, ^ is relatively small 



as compared with the corresponding values when t is near it. The curve is therefore 

 ovoid. 



