STELLAR PHOTOGRAPHY. 199 



this may be reduced to the form 

 ^ = sin S [1 — tan 8 cot (8 -\-p)'] [(1 — tan I y cot 8 cos /) cot t + tan \ y cot 8 sin €[. 



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

 8 and jy are positive, while 8 does not exceed \ tt, 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 that j/ reaches the value when 



sin |- y sin / (sin 8 cos \ y — cos 8 sin \ y cos /) = ; 



that is, when ^ = 0, when t ^z tt, 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 i tt, and at the corresponding value between | tt 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 has 



been shown that, ni the expression for -j^ , sin 8 [1 — tan 8 cot (8 + ;?)] is positive. 

 The remaining factor, since tan 8 cot ^ y = cos t, is reduced to 2 sin t cos t, 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, d ?/ is positive in both cases. 

 When t^=0, and S < i y, -^-^ = — go ; so also when t = tt, ->-, = — oo. 



Hence, when 8 < } y, the curve consists of two closed bi'anches 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 = ^ y, cos t = tan 8 cot ^ y only when ^ = ; the lower branch of the 

 curve, accordingly, disappears. The term (1 — tan ^ y cot 8 cos i) cot i becomes 



2 sin^ ^ t cot /, which may be written in the form sin \ t (cos \t — mi.\t tan \ t) ; 



d 7.1 

 when ;f = 0, this vanishes, and also the term tan \ y cot 8 sin i* ; hence ^ = 0- -A-S 



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 i tt + \ y. 



When 8 > \y, ^ = only when ?! = or when t ^ tt. When / = 0, ^ = oo ; 



when i =z TT, ^=^ — oo, as before. For values of t near 0, ^ is relatively small 



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

 ovoid. 



