Curves formed by Cephalopods and other Mollusks. 245 



to the opposite slope of the shell in those with a small apical 

 angle, and parallel to the adjacent slope when the vertical 

 angle is large, becoming parallel to the axis when 7 = 0. It 

 has also the same direction in Turrilites. 



5. To find the conditions that the whorls should intersect on 

 the same or on opposite sides of the axis. 



In equation (A) put z = r tan 7, whence 



f r _ e o cot a^ ^^(tan y cos e + sin e) 2 + (tan 7 sin e — cos e) 2 



= ±fccre ecota cosec e, 

 or 



a . f ., tea cosec e cos 7 ] 



<- sin (e + 7) s/ tc 2 + cot 2 (e + 7) > 



If the whorls touch on the same side of the axis, the larger of 

 these values must equal what the smaller becomes when + 27T 

 is written for — if they cut, greater — if out of contact, less, 

 since the similarly placed ellipses must touch if they meet on 

 the line joining their centres. The whorls therefore intersect 

 or not according as 



tea cosec e cos 7 



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



are*. / 1 _ /eg cosec e cosy 



> or <e 



I sin(e + y) v « 2 + cot 2 (€ + 7) 



as 



KG COSeC € COS y e 27rcota_^ 



}• 



sin(e + y)x//c 2 +cot 2 (e + 7) > ° F < ^ cota + l* ' ^ 



If corresponding whorls on the opposite sides of the axis 

 touch, the greater value of z when r — in equation (A) must 

 equal the smaller value of z when 6 + tt is written for 6, and r 

 again put equal to 0, since it is easily shown that if the two 

 ellipses meet on the axis they must touch. The condition for 

 this may be deduced in a similar manner to the last ; and we 

 find accordingly that shells are umbilicated or not according 

 as 



{O 2 cot 2 6 + 1) tan 7 + cot e(«; 2 — l)}(^ cota — 1) 



> or < cosec 2 6A:\/'o- 2 (« 2 cot 2 e + l) — l(e ,rcota + l). (2) 



Such an umbilicus, however, will be spiral ; a straight one, 

 caused by the generating curve not meeting the axis, will exist 

 when the radical is impossible, i. e. when 



oV/e 2 cot 2 e + l<l. 



