246 Rev. J. F. Blake on the Measurement of the 



Very few shells make the expressions above compared equal 

 to each other. Cyclostoma and ScalariaarQ examples in which 

 this nearly happens in both cases. Some Turvilitce also ap- 

 proximate to making (1) an equality, and (2) somewhat less 

 closely. 



As an example of the application of these equations, the 

 shell of Cyclostoma elegans may be taken. In this the value 

 of y, ascertained by measurement, is 78^°, e = 156°, k= '87315, 

 and a may be calculated, either directly from the equation 

 defining it or by methods to be presently noted, to be *46. Also 



14 



e 2ncotoc Dv observation = -q-. Substituting these values in 



fccr cosec e cos 7 



sin (e + 7) \/ k 2 + cot 2 (e + 7) 

 we obtain *216 approximately, while 



£.277 COt « I 



==-217 



g27rcota 1 1 <"-»-•« 



The whorls therefore, according to this calculation, are slightly 

 out of contact ; and it will be seen in the shell that the outline 

 deviates slightly from the elliptic form in order to bring them 

 into contact. Similar substitutions in the expression of (2) 

 make the left-hand side 2*679, while the left is *841. The 

 whorls therefore cut the axis, but leave a spiral umbilicus. 



6. In the case of discoid shells we need only discuss the 

 condition of contact on the same side of the axis. Here 7 = 

 and e=90°, and (1) becomes 



glir cot a __ 2 

 °"> Qr < g 2,r C o t a +1 ' 



showing that contact or overlapping is independent of the shape 

 of the whorls. Indeed it is obvious that they will be in con- 

 tact or not according as the retardation of the inner edge is 

 greater or less than 360°. As the retardation increases from 

 this value the whorls more or less overlap, becoming com- 

 pletely involute when ft is infinite, in which case a = 1. When 

 the axis is cut by the whorls, as in the case of the Nautilus, 

 whose shape is best represented by half an ellipse rapidly 

 rounded into the umbilicus, the inner edge of the major axis 

 becomes non-existent, ft being imaginary. It is by the varia- 

 tion of this element alone that such genera as Crioceras and 

 Gyroceras are separated from Ammonites and Nautilus. They 

 are therefore less distinct than Turrilites and Toxoceras, which 

 differ in addition by the whorls of the first always cutting 



