304< The Rev. H. Moseley on Conchyliometry. 



case of shells, a constant angle, represented by A, and the 

 angle P I R, being half the angle at the apex of the cone, is 

 also a constant angle in respect to that spiral, represented by *. 

 The angle P T R made by a tangent to the generating curve 

 at P with the axis of the cone is constant for the different 

 positions of the same point P on the generating curve, as the 

 curve varies its -position by the variation of 0, but variable for 

 different points on the generating curve, in any given position 

 of that curve ; let it be represented by <$>. <p is then a function 

 of the coordinates of the point P in any given position of the 

 curve, and is wholly independent of the angle which deter- 

 mines the position of the generating curve. 



Now in the right-angled spherical triangle a be, 

 cos b c = cos a b . cos a c } 

 or cos H P T = cos H P I . cos I P T, 



but IPT = PTR-RIP = $-,, 



cos H P T = cos A . cos (<$> — • i) 



.*. sin H P T = V 1 — cos 2 A . cos 2 (<p— »). 



Let R' V S' be a position of the generating curve exceed- 

 ingly near to the former, and V n a portion of another spiral 

 line on the surface of the shell exceedingly near to the spiral 

 QPw. Then may the elementary surface P V be considered 

 (in the limit) an oblique parallelogram, whose sides are 

 straight lines, and whose area is therefore represented by 



P m . P n . sin m P n. 



Let P n be represented by A s, s representing the arc R P 

 of the generating curve ; and P m by A S, S representing 

 the length of the spiral measured from the point where 

 = 0; and let it be observed that sin m P n — sin H P T 



= V 1— cos 2 A cos 2 (<J>— i) 

 .-. area P V = v' 1-cos 2 A cos 2 (<J> - i) . As.AS. 



Now the whole surface is made up of elements similar to P V* 

 therefore passing to the limit and integrating, 



whole surface =Jj \/ x _ cos s A C0g2 ^ _ ^ # dsdS 



- /TV 1 -cos 2 A cos 2 ($-i) . ~ dQ ds, (6.) 



which is a general expression for the area of a surface of re- 

 volution, whose generating curve, varying its dimensions, 

 remains always similar to itself. 



In the case of shells, if the surface of the cone on which 



