564 



THE LOGARITHMIC SPIRAL 



[CH. 



which is equivalent to 



r = 



2ab'^ cos 6 



or, ehminating the sine-function, 



_ 2ab'^ cos 6 



*' ^ (62-a2)cos2^^ 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 b, and for all values of 6, we can easily construct 

 a table, of which the following is a sample: 



Chords of an elli'pse, tvhose major and minor axes {a, b) 

 are in certain given ratios. 



The coaxial elhpses which we then draw, from the values given 

 in the table, are such as are shewn in Fig. 288 for the ratio 

 a/b = f , and in Fig. 289 for the ratio a/b = ^ ; 

 these are fair approximations to the actual 

 outhnes, and to the actual arrangement of the 

 lines of growth, in such forms as Solecurtus or 

 Cultellus, and in Telhna or Psammobia. It is 

 not difficult to introduce a constant into our 

 equation to meet the case of a shell which is 

 somewhat unsymmetrical on either side of the 

 median axis. It is a somewhat more trouble- 

 some matter, however, to bring these con- 

 figurations into relation with a "law of 

 growth," as was so easily done in the case 

 Fig. 288. of the circular figure : in other words, to 



