■_MN Rev. J. F. Blake oh t he Measurement of the 



have a curved shell (see fig. 2) in which the ornaments ap- 

 proximately run at a constant distance from the pole, while 

 the septa approximate to a radial direction. Thus one law of 

 growth is illustrated by the inside and another by the outside. 



9. The several elements of the shell have hitherto been con- 

 sidered constant; the results of their variation may now be 

 indicated. The rapid increase of a produces such shells as 

 CEkotranstes, a subgenus of Ammonites ; and the rapid diminu- 

 tion to zero, ScaphiteSj Ancyloceras, and Lituites. Its more gra- 

 dual increase with age is shown in Pupa and some Bulimi ; 

 and its diminution in many Orthocerata, which are curved in 

 youth. An increase of /3 alone produces a more involute shell, 

 as many Goniatitce and Bellerophon expansus — and in combi- 

 nation with a decrease of a, shells such as Succinea and Siga- 

 retus. Its decrease contracts the body-chambers of many 

 Orthocerata. An alteration of 7, combined in some cases with 

 an alteration of a, takes place in all those Gasteropods whose 

 whorls cannot all be touched by the same cone, and whose 

 spire may be called concave or convex. As the former is the 

 the commoner of the two, it follows that 7 most often decreases 

 with age. The changes in Vermetus and Siliquaria seem to be 

 brought about by an increase in 7 alone. Its rapid decrease, 

 on the contrary, bringing it to a negative value, produces the 

 depressed spires of some Cones and of the Cyprcew. When 

 all three vary simultaneously, the effects may be so masked as 

 to be ascertainable only by careful measurement. 



In the Cephalopoda, which have part of their shells parti- 

 tioned off, some authors give among the characters both the 

 length and the capacity of the body-chamber. These two are 

 evidently deducible from each other, whatever may be the 

 shape of the shell, provided its growth continues constant; 

 for if the length of the body-chamber =(1 — k) whole length, 

 the capacity = (1— /c 3 ) whole capacity : if therefore it can be 

 measured independently, it will show if any contraction or 

 expansion has taken place. 



It remains now to show how the three elements of shells 

 may best be observed, and especially on imperfect specimens 

 such as fossils very commonly are. 



10. To find the spiral angle of a discoid or turbinated shell. 

 Since in a turbinated shell all the radii from the apex and 



arcs of the curve lying on one cone are multiples of their pro- 

 jections on the plane perpendicular to the axis, we need only 

 consider discoid forms at first. Let ABCD (fig. 3) be such a 

 form ; AC, BD two diameters at right angles through its pole. 

 Then, from the properties of the equiangular spiral, it is obvious 

 that we have a choice of several methods of determining a, by 





