traced upon the Surface of the Sphere. 395 



guments at length ; but proceed at once to apply them to the peculiar ob- 

 jects before us, that is, for obtaining the formulae adapted to spherical co- 

 ordinates. These formulae, it is at once obvious, have an intimate relation 

 with those in piano, and often embrace them as particular cases : never- 

 theless the spherical can never be inferred from the plane, though the plane 

 can most commonly (still not always so) be inferred from the spherical ex- 

 pressions. 



2. But we may, as is always done in the investigation of spirals in piano, 

 employ the equation between the radius-vector and perpendicular upon 

 the tangent. 



Considering 4 and ^' as functions of <p and 0' respectively, we shall arrive 

 by similar considerations to those before used, at the conditions 

 , d+ d# dt *</ 



*=*'*=* 10= -' '-'-'- 



as the test of contact of the (w+lth) order, and all the consequences usually 

 drawn will follow from these, as well as from the previous ones. 



XLIV. 



Radius of Spherical Curvature, Involutes and Evolutes. 



By the radius of spherical curvature, I mean the spherical radius of the 

 circle which has a contact of the second order with the given curve, at the 

 current point (p Q, or (p ^, according as we use one or other of these pairs of 

 co-ordinates. 



1st, We shall first take the equation between ^ (p. 



Here F(0'4/) = (1.) 



, ,, . , ,. , /VT -VT , ,, v cosd)cosp cosX ,~, 



and the circle of contact (XLV. 1 5.) is sm 4- = dz r-^-r ( 2 -) 



sm<p 



Now the equation (2) involving the two arbitrary constants p and X, these 

 may be so determined as to fulfil the conditions 



, d$ 



sm V = sin y and = 

 d(p 



VOL. XII. PART II. 3 E 



