356 Mr DAVIES on the Equations of Loci 



find, then, the number of points which are furnished of the locus or, in other 

 words, of how many branches the locus may at the most consist we must resolve 

 all the functions of <p into one particular function (no matter which) of <p, and all the 

 functions of 6 into any particular function of 6. If the degree of this polynome be 

 , then the number of points furnished by each value of the trigonometrical function 

 which is taken as the principal variable, will determine^wr points on the surface of 

 the sphere. The curve may, then, be composed of 4n branches: at least in an ana- 

 lytical point of view, such will be the case. If, however, we seek the true geometri- 

 cal number, we shall find it to be 2w, as each of the pairs which result from this re- 

 solution, are reduplications of the other pair, each of each. Either branch may be 

 described by a diameter of the sphere moving upon the other branch, and leaving 

 upon the sphere the trace of its other extremity. This second branch is, indeed, the 

 locus of the second intersection of the circles whose primary ones trace out the pri- 

 mary branch of the locus, when the locus is that of the intersections of two circles, 

 described according to an assigned law ; and, in all cases, there is some circum- 

 stance in the signification of the problem correspondent to the second intersection. 

 We have only to lay down the relations of these branches, as they depend upon the 

 functions by which they are determined. 



1 mo, Let it be resolved for (p. Then taking the polar equation, 

 sin? =/() (5.) 



This denotes two points upon the same meridian, equidistant from the poles, and 

 the locus itself surrounding both poles, is composed of branches perfectly equal and 

 similar in all respects, except that these two branches are also upon a right cylinder, 

 which is perpendicular to the equator. The primary and secondary branches are 

 therefore cylindrlcally related to each other. 



Next, take cosip=/(0 (6.) 



This gives two branches, lying opposite ways upon the radius vector (p ; these may 

 become coincident ; they do so when the value of cos <p is single, and marked + 

 or . 



Again, take tanip = /() (7.) 



Now, since tan = tan (nr + (p), there are two points belonging to this diametri- 

 cally opposite to one another upon the sphere, each of which may be considered to 

 trace a branch round either pole. These, as before, may become coincident. The 

 cotangent also being the reciprocal of the tangent, the same circumstance takes 

 place. The equation of the hour-lines on the Antique Dials is an instance of this. 

 If we consider the polar equation of these curves, viz. cot(f> = +tanAcosnd, 



