traced upon the Surface of the Sphere. 



301 



But tan ^ D = i- d A, in the case of elementary triangles, and hence we 

 have ultimately 



A = f (I cos<p)dO + const. * 



XVII. 

 THE EQUABLE SPHERICAL SPIRAL. 



We shall now proceed to the application of these principles to spheri- 

 cal loci, different from those (viz. the circles) which we have been hitherto 

 considering. It would be easy to imagine different methods by which curves 

 may be traced as well on the sphere as in piano ; but it will perhaps be 

 more agreeable, and at the same time better display the advantages of the 

 method here employed, if we examine sphericals already imagined, and which 

 have been often treated by other methods* We shall therefore commence 

 by considering the SPIRAL OF PAPPUS f . We may state it more generally 

 thus: 



A meridian PHP' revolves about the axis PP' of a sphere, whilst a 

 point M in it moves from P in the direction of PRP' ; these motions be- 

 ing uniform, and in a given ratio. What is the locus of M ? 



* Certain precautions, which are not very prominently brought forward in piano, 

 are necessary in the use of this and of all other forms of spherical equations. We 

 shall explain at a future time. 



f PAPPUS, Coll. Math. lib. iv. prop. 30. 

 VOL. XII. PART I. O Q 



