THE SHELLS OF MOLLUSCS 



157 



D'Arcy Thompson (1. c.) has treated the quantitative aspect 

 of this problem at greater length. Unfortunately, he has not 

 analysed the whole process, as would biologically be the ideal 

 method, in terms of two co-operating growth-ratios, but, as 

 regards part of the problem, he has been content to give a 

 purely mathematical description. An analysis entirely in 

 terms of growth-ratios is being undertaken by Professor H. 

 Levy, of the Imperial College of Science ; meanwhile it will 

 be useful to give a brief summary of D'Arcy Thompson's 

 treatment of the matter. 



The form of a single curve following a logarithmic spiral 

 is given by the expression 



.,, _ pQ COt a 



where r is the radius of the shell from centre to circumfer- 

 ence ; 6 is the angle of revolution which the spiral has de- 

 scribed ; and a is the angle between the tangent of the curve 

 and the radius vector of the curve, which remains constant : this 

 is known as the constant angle of the curve. In addition, the 

 ratio of the radii of successive whorls is also always a constant. 

 The relation between these two constants is given in the 

 following table (abbreviated from D'Arcy Thompson, p. 534) : 



Ratio of breadth 



of each whorl to 



the next preceding 



1-0 



1-5 

 2-0 



3-o 



5-o 



io-o 



50-0 



ioo-o 



10,000 



1,000,000 



100,000,000 



In Nautilus, in all 

 shells of Gastropods, 

 usually between 8o° 

 of successive whorls 

 decreasing values of 

 out very rapidly. E 

 the first whorl were 



16 52' 



ordinary shells, and in all typical spiral 

 the constant angle is rarely below 8o°, 

 and 85 , and the ratio of the breadth 

 usually between 3-0 and 175. With 

 the constant angle, the spiral flattens 

 .g., if the constant angle were 28 , and 

 1 in. broad, the next whorl would be 



