826 THE EQUIANGULAR SPIRAL . [ch. 



That is to say, all the balls reach the circumference of the 

 circle at the same moment as the ball which drops vertically from 

 ^ toC. 



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 faUing 

 body. And, mutatis mutandis, a similar definite law underlies the 

 cases where the generating curve is continually elliptical, 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^/a^ + y^jb^ = 1, where a and b are the major and minor 

 axes of the ellipse. 



Or, changing the origin to the vertex of the figure, 



x^ 2aj 1/2 ^ . . (x— a)^ y^ 



Then, transferring to polar coordinates, where r . cos 6 = x, 

 r . sin 6 = y, we have 



r . cos^ 6 2 cos 6 r . sin^ 



which is equivalent to 



2a62 cos d 



r = 



b^cos^d + a^sin^d' 

 or, simpHfying, by eliminating the sine-function, 



2ab^ cos d 



r = 



(62_a2)cos2<9 + a2* 



Obviously, in the case when a = b, this gives us the circular 

 system which we have already considered. For other values, or 

 ratios, of a and 6, and for all values of 6, we can easily construct 

 a table, of which the following is a sample : 



