The Rev. H. Moseley on Conchyliometry . 301 



whose apex is 2 , the pole of the spiral coinciding with the apex 



of the cone. The circular arc 8, whose radius is unity when 

 developed, will when wrapped upon the cone, become a cir- 

 cular arc 0, whose radius is sin i, 



.*. 8 = sin t, 

 whence it follows that R representing the distance of any 

 point in the spiral from the apex of the cone, and the angle 

 included between two planes, intersecting in the axis of the 

 cone, one passing through that point of the spiral, and the 

 other through the point where R = R , we have 

 R = R s<> sin ' cotA . 

 Let Ri R 2 R 3 , &c. be distances from the apex of the cone 

 of points of the spiral in the same straight line passing through 

 the apex, 



Q representing the quotient of any two consecutive distances 

 between the whorls measured on the same straight line passing 

 through the apex. 



On the supposition made therefore, viz. that a plane lo- 

 garithmic spiral is wrapped upon a cone, its pole coinciding 

 with the apex of the cone, it follows that the distances of the 

 successive whorls of the spiral measured on the same straight 

 line passing through the apex of the cone, are in geometrical 

 progression ; and conversely. Now in shells they are found, 

 by admeasurement, thus to be in geometrical progression. 

 The spirals described on shells, and called concho-spirals, are 

 therefore such as would result from winding plane logarithmic 

 spirals on cones. 



To determine in respect to any shell the constant angle A 

 which the tangent to its concho-spiral when developed makes 

 with its radius vector, let it be observed that 

 log s Q = 2 7T sin » cot A 



. 2 7rsin i ,. . 



.•. tan A = -j p^, (1.) 



log, Q 



where A is the angle required. 



