402 Mr DAVIES on the Equations of Loci 



less general than (xiv. 5.), it is possible to annex some other condition (one 

 or more, as subsequent investigation may bhov, to _be .admissible), to those 

 involved in bur present article ; or, in other words, that the section of the 

 cone by a sphere under some less confined relation may still give such a 

 spherical conic section as \ve have previously defined. Comparing the pre- 

 sent with the case where the sphere becomes a plane, we see the one case is 

 given by a concentric sphere and the other a sphere v.-hose centre is infinitely 

 distant. It is natural to ask, if the analogy is confined to these two cases, 

 or may the centre of the cutting sphere be situated any where in space ? 

 We have not room to enter upon this inquiry ; but we may just state that 

 the last is not the case. It is in one specific point, determined by the mag- 

 nitude of the constant on the left side of the equation. 



XLVI. 



It will readily appear, that by taking the radius of the sphere infinite, the 

 modifications which the spherical equations undergo, ought to have some 

 analogy to the equations of plane curves. We cannot, it is true, always in- 

 fer the plane from the spherical equation, much less the spherical from the 

 plane, when there is an analogy between the genesis. The discussion of 

 this curious question is omitted here, not as irrelevant, but as too long to 

 be admitted on the present occasion. There is, however, in general, a consi- 

 derable degree of analogy between the two ; and to shew this in perhaps the 

 most important case that can occur, I shall investigate, ab initio, the polar 

 equation of a straight line, for the purpose of comparison with the polar 

 equation of a great circle given in my former paper. 



On the Polar Equation of a Straight Line. 



1. The equation of a straight line, the origin of the polar angle being 

 OC, and the origin of the radius-vector being O, is thus found : 



