XI] 



OF BIVALVE SHELLS 



563 



all these moving bodies is a circle passing through the point of 

 origin. For the acceleration along any one of the sloping paths, 

 for instance AB (Fig. 287), is such 

 that 



AB = \g cos e . t'^ 



= Ig . ABjAC . f\ 



Therefore 



• e- = 2lg.AC. 



That is to say, all the balls 

 reach the circumference of the 

 circle at the same moment as the 

 ball which drops vertically from 

 A to C. 



Where, then, as often happens, the generating curve of the 

 shell is approximately a circle passing through the point of origin, 

 we may consider the acceleration of growth along various radiants 

 to be governed by a simple mathematical law, closely akin to 

 that simple law of acceleration which governs the movements of 

 a falling body. And, mutatis mutandis, a similar definite law 

 underlies the cases where the generating curve is continually 

 elhptical, or where it assumes some more complex, but still regular 

 and constant form. 



It is easy to extend the proposition to the particular case where 

 the lines of growth may be considered elliptical. In such a case 

 we have x^la- + y'^jh^ = 1, where a and h are the major and minor 

 axes of the ellipse. 



Or, changing the origin to the vertex of the figure 



2x 



giving 



r 



t 

 62 



= 0, 



= 1. 



Then, transferring to polar coordinates, where r . cos d = x, 

 r . sin 6 = y, we have 



r . cos^ d 2 cos dr. sin 

 a^ a ' h^ 



= 0, 



36—2 



