S02 The Rev. H. Moseley on Conchyliometry. 



Now the quotient Q is the same for all the spirals described 

 on the surface of the same shell ; if then we represent 



logs Q , 

 * bye, 



2 7T * 



we have sin i cot A = — ° = c, 



2 7T 



and R = n B c ° (2.) 



which is the general equation to a concho- spiral. 



Since each of the concho-spirals on any shell must have 

 its origin in a corresponding point of the generating curve of 

 that shell when in its initial position, and since the initial 

 dimensions of the generating curve of every such shell are ex- 

 ceedingly (perhaps infinitely) small, it follows that all such 

 spirals have their origins very nearly (perhaps accurately) in 

 the same point, and therefore that the conical surfaces on 

 which they are severally described have their apices in the 

 same point* ; the value of R being the distance from the com- 

 mon apex to that particular point of the generating curve, at 

 which the spiral intersects it, in that position in which is 

 assumed to be zero. 



II. To determine the inclination u of the tangent at any point 

 of a concho-spiral to a line drawn from that point parallel to 

 the axis of the shell. 



Let P Q represent any portion of a concho-spiral, P H a 

 tangent to it at P, P L a line drawn from P parallel to the 

 axis I R of the shell, I the apex of the cone on which the 

 concho-spiral is described. Join I P, then is I P H a con- 



stant angle represented by A, and H P L (represented by «) 

 is the angle required. 



Describe a sphere with radius unity from the centre P, and 

 let a ?, a b, b e be the intersection of the planes I P L, I P H, 

 L P H with its surface. The spherical angle bae is a right 



* It is a law common to all surfaces of revolution whose generating 

 curves varying their dimensions remain always geometrically similar, that 

 the spiral lines described by given points in these curves lie all on the sur- 

 faces of cones having a common apex. 



