traced upon the Surface of the Sphere. 357 



we shall find a perfect correspondence between this result and that given in the pa- 

 per on that subject. The two branches are here canically opposite. 



In case of secant and cosecant of (p, we have the same class of results as those 

 afforded by the cosines and sines, of which they are the reciprocals. 



%do, Resolve the equation for <p : and take it as a rectangular equation. We 

 shall have simply an interchange of sines with cosines, tangents with cotangents, 

 and secants with cosecants. The sines will now give cylindrically opposed curves, 

 and cosines reduplicating curves : tangents and cotangents as before : and secants 

 and cosecants give results similar in that respect with the cosines and sines. 



3tio, Resolve it for 6 : it will now be more convenient, generally, to consider 

 the curve as referred to rectangular co-ordinates. 



The sines will give curves cylindrically opposite, and referred to the prime meri- 

 dian and its opposite branch. The cosines will give curves symmetrically related to 

 prime meridian, lying to the right and left of it. The tangents and cotangents will 

 give curves of branches diametrically opposite ; and the secants and cosecants, the 

 same classes as those furnished by cosines and sines. 



NOTE C. 



THOUGH I have limited myself to a mere indication of the general character of 

 the spherical conic sections, it may not be irrelevant to state here, that all the pro- 

 perties of contact and intersection, all the properties of inscribed and circumscribed 

 polygons, and most of the loci generated from these curves, by given methods, when 

 the figures are taken in piano, have analogues on the surface of the sphere. The 

 circle, however, which in the plane conic sections becomes a locus in consequence of 

 containing a given angle subtended by a base given in magnitude and position, is 

 exchanged in sphcero for the curve discussed in XXXVI and XXXVII: but this 

 change does not take place when the locus is the path of a given angle contained by 

 circles described by any other law. The perpendicular from the focus upon the 

 tangent is an instance of this latter kind. In piano and in splicero, it is alike a 

 circle on the major axis. The various divisions of great circles any way related to 

 the defining data, or to lines definitely drawn with respect to the fixed lines and 

 fixed points of the locus, have analogues in terms of some of their trigonometrical 

 functions to the sections of the corresponding straight lines in piano. A few of 

 VOL. XII. PART 1. Z Z 



