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ROYAL SOCIETY OF CANADA 



initial position. This gives us tan -^ = Ke, where K is written 



in place of a function of 0' r' and c, which is readily calculated. 

 This curve, then, is the analogue in spherical analytical geometry of 

 the equiangular spiral in plane. In fact if the radius of the sphere 

 were infinite we should have an equiangular spiral about the origin. 

 XXXVII. The centre of curvature for any point P on a curve 

 is defined to be the limiting position of the intersection of normals at 

 P and Q where Q is another point on the curve, as Q moves up to P 

 along the curve. The radius of curva- 

 ture will be the spherical radius of the 

 small circle passing through P with the 

 centre of curvature as centre. It will be 

 seen that there will be two centres of 

 curvature and two radii of curvature; 

 the sum of the radii will be tz. Further- 

 more, the maximum radius of curva- 



ture is — . Suppose P (3 a secant of 



the curve and let the normals of length ^o ^ il 



meet at C. Then 



cos C = cos r cos |0 -f- sin r . sin /) sin ^ 



= cos (r + A ^) cos |0 -f- sin (r + /\r) sin p sin {({) + A (f>) 

 That is on simplification 



Ars'mr + ( ( A ^') ) 



tan /J = -^ r cos r sin -1- A sin r cos ^ + ( (A ?•')) + ( (A <!>')) 

 Now the limit of the left side exists through the circumstance that the 

 right side has a definite limit. Calling p the radius of curvature, then 



sin r 

 tan p 



cos r sin (p + 



dip 

 dr 



sin r cos 



1 



cot p = cos r 



cot *• sin + 

 dd 



d s 



+ 



dj}_ 

 dr 



dip 



d s 



cos 



Or 



dd , dip. . 



tan p (cos r —J— + —j — ; = i. 

 ^ ^ d s d s 



The result shews that {p - n) answers as well as p, which we pre- 

 dicted. The result also gives the customary formula in plane geo- 

 metry that 



d<f> 



ds 



= 1. 



