308 



Mr DAVIES on the Equations of Loci 



XX. 







The identity of the solutions of VIVIANI and BERNOULLI of the Flo- 

 rentine Enigma, was asserted in (XIX, 2) : we shall prove it here. 



Fig. 14. 



Let a plane perpendicular to the axis cut the cylinder in the curve 

 NGH ; and let a diametrical plane, perpendicular to the plane NGH, cut 

 the sphere in PQ, and the plane NGH in OQ. Take the pole of NGH 

 for origin of < ; and let the equation of the generating curve NGH (refer- 

 red to pole O, and polar angle QON) \>ef(Q, v). Draw MN _L to the plane 

 NHG, then MN is one of the edges of the cylinder, and M a point in the 

 curve of penetration. Draw MR parallel to NO. 



Then RM = ON = v = a sin = sin <p (rad. = 1). 



Hence the equation of the curve of penetration of the sphere and any cy- 

 linder is 



/(0,sin(/>) = 0* ............................. (1.) 



But in the circle on OQ, to which the present question refers, we have 

 v = cos Q cos Q to rad. 1. Insert this in (1). 



Hence the equation of the curve becomes, (see fig. at top of next page), 



sn ( = 



* For MPQ = NOQ = 6 ; or is the same both on the spherical equator and its 

 projection on the plane MHG. It may also be remarked, that there is no essential dif- 

 ference between BERNOULLI'S own proof and the one above given, except the notation. 



