330 Mr DAVIES on the Equations of Loci 



(that is, it is a transcendent in the same cases as the plane curve), and yet 

 has the unexpected property of being rectifiable : but it is by no means a 

 singular case, for innumerable curves may be traced on a sphere combining 

 the two properties. The general condition of rectification, indeed, involves 

 an arbitrary function, whichever of the variables be eliminated ; and hence 

 it is easy to see that the property must be a very extensive one. I cannot, 

 however, enter into details here, but shall reserve what I have- to say on this 

 topic till a future occasion. 



Note. The most scientific geometrical view of the spherical epicycloid 

 has been given by HACHETTE, in the Correspondence of the Polytechnic 

 School, vol. ii. p. 24. ; in his supplement to the Descriptive Geometry of 

 MONGE, p. 88. ; and in his own treatise on the same subject, p. 159. It 

 might also be mentioned, that CLAIRAULT* has exhibited quadrable por- 

 tions of the spherical epicycloid ; and that, did our limits permit, we might 

 effect the same very elegantly by our own methods. It is not necessary 

 here, however, to discuss that question. 



XXVIII. 



THE SUN'S VERTICAL PATH OVER THE EARTH. 



AN elegant problem was proposed by JOHN BERNOULLI in 1732, 

 which has a close analogy to this inquiry, a solution of which we shall set 

 down. 



What is the path of the vertical projection of the sun upon the earth's 

 surface, supposing the sun to move round the earth in a circle with an 

 equable motion, and the earth itself to be a sphere ? 



sine of the declination as sine declination is to radius. Making these transformations, the 

 result I have given above will be found identical with BERNOULLI'S, though obtained by 

 so completely different a process. 



* Mem. de 1'Acad. des Scien. 1732, pp. 293-4. 



