MACALISTEB. ON THE GROWTH OF TURBINATED SHELLS. 



23 



Hitherto we have been examining the formulas for discoid shells, but 

 by far the greater number of shell-forms are those in which the whorls, 

 instead of remaining in the same plane, slide down on the central axis, 

 thus making a turbinated shell-form. A new principle enters into our 

 calculation here ; for the shape of a turbinated shell depends on the 

 mutual relation of three, and not two constants. These are, first, the 

 form of the generating figure ; secondly, the discoidal coefficient m ; 

 thirdly, the helicoidal coefficient n. Upon the relations of these para- 

 meters to each other depends the shape of the shell. Thus in some n 

 is nearly equal to m, and in such cases the whorls scarcely embrace 

 each other, and the figure produced is that of an elongated cone, as in 

 the genera Turritella, Cerithium, Acus, &c. Sometimes n exceeds m ; 

 and in this case the resulting form is an open spiral as in Vermetus, or 

 a rapidly descending series of whorls. A third possible case is that in 

 which n is less than m, and the resulting figure is globular ; but of this 

 case, though a possible one, I have not as yet succeeded in obtaining an 

 example. 



The following cases illustrate the formula n> m : — 



Vermetus lumbricalis, 

 Delphinulaatra, . 



1-42 

 6-00 



1-3 



2-85 



Width oi Whorls in 

 decimals of an inch. 



0-075 

 0-018 



0-1 



0-5 



0-13 

 0-148 



0-175 

 0-41 



Amount of Translation. 



0-15 

 001 



0-22 

 0-05 



0-3 0-45. 

 0-3 



The following instances exemplify the case n = m 



Species. 



n=m. 



Length of Whorls in decimals of an inch. 



Helicostyla polychroa, 



2 



0-41 



0-081 



0-158 



0-32 



0-7 

















Fusus colosseus, . . 



1-71 



0-09 



0-14 



0-26 



0-43 



0-76 

















Ph asianella bulimoides, 



1-8 



0-07 



0-125 



0-23 



0-45 



















Scalaria preciosa, . . 



1-56 



0-05 



0-078 



0-13 



0-2 



0-32 



0-52 















Fusus antiquus, . . 



1-5 



0-15 



0-225 



0-343 



0-54 



0-84 

















Mitra episcopalis, . . 



1-434 



0-245 



0-4 



0-575 



0-82 



















Trochus niloticus, . . 



1-41 



0-2 



0-3 



0-425 



0-63 



0-9 



1-2 















Fusus longissimus, . 



1-341 



0-25 



0-3 



0-44 



0-6 



0-81 

















Fusus colus, . . . 



1-33 



0-15 



0-2 



0-26 



0-35 



0-42 



0-54 



0-83 













Pyrazus sulcatus, . . 



1-33 



0-13 



0-17 



0-29 



0-38 



0-51 

















Acus dimidiata, . . 



1-277 



0-2 



0-267 



0-31 



0-4 



0-52 



0-62 



0-88 













Acus maculata, . . 



1-25 



0-15 



0-176 



0-23 



0-29 



0-37 



0-45 



0-53 



0-7 



0-9 









Acus crenulatus, . . 



1-25 



0-2 



0-35 



0-32 



0-38 



0-496 



0-6 















Cerithium nodulosum, 



1-24 



0-23 



0-3 



0-37 





















Pirena terebralis, . . 



1-23 



0-08 



0-12 



0-15 



0-178 



0-22 



0-28 



0-35 













Pyrazus palustris, 



1-22 



0-15 



0-182 



0-22 



0-27 



0-34 



0-42 



0-5 













Zaria duplicata, . . 



1-23 



0-078 



0-1 



0-125 



0-16 



0-2 



0-24 



0-3 



0-26 



0-44 



0-53 



0-625 



0-76 



Acus subulata, . . . 



1-163 



0-175 



0-2 



0-23 



0-265 



0-32 



0-367 



0-432 



0-47 



0-641 









Telescopium fuscum, 



1-14 



0-1 



0-112 



0-125 



0-15 



0-18 



0-2 



0-24 



0-28 



0-325 



0-365 







The Eev. Dr. Haughton read a Paper " On the Geometrical Classi- 

 fication of Muscles." Eeferring to Borrelli's old classification he made 

 several alterations in it, adding to it, among others, the class of muscles 



whose 

 faces. 



fibres, though straight lines, yet conjointly make skew sur- 



