Curves formed by Cephalopods and other Mollusks. 259 

 To find M(e<£), put cos <f>=a& ; 



.-. J \/l — e 2 sin $ sin <f> d<p = — jV 1 — e 2 + ¥s?dx= — ej-y/ — ^-+^ 



= -|{^ v/l-e 2 + eV+^^log^+^Vl-e 2 + eV)|. 



When 



cf> = 0, x=l; 



■■■ M(e«=l{U^log(l + ^)- C -^VW^^ 



— log(cos<£ + - \/l — e 2 sin 2 <£] J- 



, /-, , ,^ o.o. x 1 — e 2 , ecos6+ \/l— e 2 sin 2 (f>. 



= i.(l-cos0va-e 2 sm 2 <£) ^- log 1+e r ' 



consequently 



da 



M 



( e l)**~if l0 «r+|- 



It now remains to determine cr and c£ in terms of R and A, 

 and 0' in terms of 6. If (/>, <f> / be the complements of PBN, 

 PAN respectively, we have from the figure (fig. 9), 



crsin# = (l-X-X*r)(=gD, .... (2) 



K^sm</)+sin(/) / = — — (=£^J, . . (3) 



R 2 cos</>=cos<jf>Y=^g-j; (4) 



from these we obtain 



a==1 ~w+i (5) 



By the compression of the shell the radii, and therefore the #'s 

 corresponding to them, will be enlarged ; and this will take 

 place so that the sum of the arcs of the ellipses in one plane 

 may be constant. This gives 



^,~*{^) + ™«$^™ }, (6) 



whence, making the necessary substitutions in (1), we obtain 

 the condition for identity. 



S2 



