Curves formed by Cephalopods and other Mollusks. 257 

 whence the required relation deduced from the equality of the 



areas is 



i 



I V sec 2 co -e 2 (l + tan 2 co sin 2 0) rl . 

 sec 2 &) — e 2 cos 2 </> 



= H ^c 2 ft) / -e /2 (l + tan 2 ft/ sin <j>) rl , 

 J sec 2 ft> / — e /2 cos 2 ^> 



The impossibility of exact measurements on fossil shells 

 which have been compressed renders the working out of these 

 integrals, representing the correct solution of the question, a 

 waste of time, if a fair approximation may be made some other 

 way. No very close approximation which is practicable to 

 work has been found. If, for example, we take for the length 

 of the arc the sum of the chords subtending J the arc of a 

 circle passing through the ends of the major axis and the sphe- 

 rical projection of the minor axis, and if © l3 co 2 be the angles 

 subtended at the centre by the semiaxes, then 



tan G> 2 



k= J 



tan g)! 



and 



radius of circle 



A(l — cos ft)j cos ft> 2 ) 

 s/ 1 — 2 COS ft>! cos &> 2 + cos' 2 &>i 



whence 



n COS ft>! (COS ft) 2 — COS ft>i) 



COS C7= = — J 



1 — COS &>! COS ft> 2 



. 6_ - /\/2(l — cos ft>! cos ft> 2 ) — sinft?! 

 4 v 2^2(1—008 &>! cos ft) 2 ) 



and if the chords be assumed constant, we have 



p__ (1 — cos ft>! cos &) 2 ) 2 v?2(l — cosft> 1 cosft) 2 ) — sin o)i 



" 2(1— cos ft) x cos ft) 2 ) — sin 2 ft) x ' V 1 — cos co x cos co 2 



or 



(1 — COS ft)! COS ft) 2 ) ¥ (1 — COS ft) / iCOS &/ 2 ) ¥ 



\/ 2(1 — cos ft)iCOS &) 2 ) + sin co\ \/ 2(1 — cosew/cosft^) 4- sina/i 



If this relation is satisfied between two shells, one may be the 

 compressed form of the other. 



For example, a specimen of OrtJwceras annulatus has its 

 minor axis decrease from 20 to 16 lines in 38, the major axis 

 at the larger size being 32 ; to compare this with McCoy's 

 description "tapering 1 in 8," the uncompressed section being 



Phil Mag. S. 5. Vol. 6. No. 37. Oct. 1878. S 



