332 Mr DAVIES on the Equations of Loci 



n i / COS(i\ 



6 n % = n cos- 1 ( ,-- ] , or 

 V smX / 



cosd) = sin Xcos - 6 . .(5.) 



n 



The equation of the curve is thus found to be very simple : and it may 



7T 



be further remarked, that if X = g, it becomes the equable spherical spiral 

 already considered. For then 



sin X = 1, and - 6 = cos" 1 ( cos <p) = ir (b, 



n. x ' * 7 



n 



which differs from the equations already found only in being referred to a 

 different origin either of < or 6. We may consider it to refer to (j> mea- 

 sured from the opposite pole, or to 6 measured from the opposite meridian. 

 In our equation of the circle, too, the same result is indicated, and we might 

 therefore look for it here. 



The equation (5) given above may, indeed, be considered as the most 

 general form of the equable spherical spiral, and from which all the parti- 

 cular cases (as the spiral of PAPPUS, the oval window of VIVIANI, &c.) may 

 be at once derived. The discussion of the question under this aspect might 

 be interesting, but, after the detail into which we have already entered, it 

 is not necessary to do it here. 



XXIX. 



The proposal of this curve, however, originated in a mistaken analogy 

 between this combination of motion and that which produces the spherical 

 epicycloid*. About the same time, MAUPEimusf proposed and offered 

 a solution to a problem having some similarity to this, viz. to find that me- 

 ridian on the celestial sphere, where the motion in right ascension and in 

 longitude have a given ratio to one another ; and it had been discussed al- 



* Mem. de 1'Acad. 1732, p. 245 ; or Opera Omnia, torn. iii. p. 226. 

 f Mem. de 1'Acad. 1732, pp. 257-8. 



