traced upon the Surface of the Sphere. #53 



There are hence on each side of the circle RT, two branches of the 

 curve, which are two and two symmetrical. They intersect PQ PQ' in 

 points MN, M'N' ; so that PM = PM', and PN = PN'. To ascertain 

 whether the symmetrical branches are likewise continuous or not, we have 

 only to consider that the same value of gives unequal values of sin <p, and 

 hence the two continuous pairs of branches are not the symmetrical ones. 

 The curve, then, is composed of two separate ovals, which intersect each 

 other in R and T ; but they are at the same time in all respects equal, and 

 only situated in a reversed position, with respect to each other and RT, 

 as in the annexed figure. The two unequal branches into which RT di- 

 vides each oval, are the loci of the given angle, and of its supplement ; as is 

 obvious from the formula determining the locus by means of the sine of the 

 vertical angle, that sine being also the sine of the supplement of the given 

 angle. The same thing takes place when the problem is solved in piano : 

 the locus by these two circles very much resembling our figure above in ge- 

 neral appearance, and having several analogous properties. 



Besides these, there are two others, which constitute the locus of the se- 

 condary intersections of the circles RM, TM, which contain the given angle, 

 and surrounding the opposite pole. The secondary branch of RNP is co- 

 nically * opposite to its primary, and cyliudrically opposite to the primary 

 branch of RN'T : and, vice versa, the secondary branch of RN'T is coni- 

 cally opposite to its primary, and cylindrically to the primary branch of 

 RNT. The primary and secondary branches of each individual (both of 

 RNT and RN'T) are obliquely cylindrical to each other : and so are the 

 two primary branches themselves, as well as their secondaries, obliquely 

 cylindrical. 



Vide Note B. 



