upon the Antique Dials. 103 



ference will be found between them ; the one involves the secant of the vari- 

 able angle 9, and the other involves the square of the secant of half the 

 angle 0. 



In the figures to this and the 

 last case, FG is the projection of 

 the equator upon the cone, QR 

 the parallel of declination, which 

 bounds the hectemoria on the he- 

 misphere, which is touched by the 

 cone. The opposite boimdary in 

 figure to Case 1. is Q'R' ; but in the second case, it is infinitely distant. 



3. Let cot x cot I ^ +1. Then put cos r = cot * cot I. 



When wzL = 180 v, then the denominator vanishes, and the radius- 

 vector r becomes infinite. The curve has at this value of L an asymptote. 



During the onward motion of mJj the denominator becomes negative, 

 whilst the value of r diminishes in value till wL = 180, at which point 

 the curve bends round, and forms an equal branch running off to infinity ; 

 which it attains when ?L = 180 + v, and the curve has another asymp- 

 tote. After this the denominator becomes positive, and continues so till 



c; and it proceeds through the same system of changes 

 as before. 



Thus, if SPT be the branch cor- 

 responding to the values between 

 1 80 + i and 380 x as values of m L, 

 then S'P'T' will be that correspond- 

 ing to the remaining portion of the 

 circuit lying between 180 r and 

 180 + K. The same alternation of op- 

 . posite curves will succeed each other 

 at regular intervals, as in the figure. 



o 2 



