traced upon the Surface of the Sphere. 305 



1. When (f> = 6, we have 



2. When < = j, we have = 4, or Tra = 4 and n = \. Hence, the length 



T? 71 



of the spiral of PAPPUS is 



-rr-k'U- 1 



It is unnecessary to pursue these rectifications farther, as the character 

 of the inquiry is well known to geometers. 







XIX. 



The areas of these spirals can in some cases be expressed by means of 



spherical lunes. 



The general expression for the element of the arc is 



, . in 



' n 

 m 



' (I-) 



or A = (<p sin (p) + const (2 ) 



H 



1. Let m = n. Then taking the integral between and -jr, we have 

 for one-fourth of the area of the curve, 



or for the whole area, A 2 = TT 4. The radius in all these cases being 

 unity. 



The residue of the hemisphere is 



2 ir (2 v 4) = 4, 



which being equal to the square of the diameter of the sphere, that resi- 

 due is quadruple. 



