OF TURBINATED AND DISCOID SHELLS. 



353 



The following distances were measured upon three different opercula from the poles 

 of their spiral curves to their successive whorls ; the distances in the same column 

 being measured on the same radius vector produced. It will be perceived that for 

 the same operculum these distances have the same ratio consecutively to one another; 

 the deviation from this law in no case exceeding that error which of necessity 

 attaches to the method of admeasurement. 



Operculum, No. II. 



Operculum, No. III. 



The spiral of the operculum is then a logarithmic spiral. Now its linear dimen- 

 sions in the different successive stages of its progress have been shown to be as the 

 successive radii vectores of its spiral. The increments of its linear dimensions are 

 then as the increments of these radii vectores. But by a fundamental property of the 

 logarithmic spiral, the increments of its radii vectores, corresponding to equal incre- 

 ments in their angles of revolution, are as the radii vectores themselves. Thus, then, 

 it follows that the increments of the linear dimensions of the operculum, correspond- 

 ing to equal angular distances round its pole, are as its existing linear dimensions ; 

 and, therefore, that the increments of the linear dimensions of the section of the spi- 

 ral chamber corresponding to these are everywhere as its existing linear dimensions. 



The animal, as he advances in the construction of his shell, increases continually 

 his operculum, so as to adjust it to its mouth. 



He increases it, however, not by additions made at the same time all round its 



MDCCCXXXVIII. 2 z 



