Curves formed by Cephalopods and other Mollusks. 255 

 d=12 millims. and a=3*7 ; also 



«=-87315 ; e = 156°, R=y/M. 



.-. 12 x 2-4586<r = 3-7 x 1-80178{1 +0-2-2216} ; 



.-. 14-6926 o-= 6-6666, and o-= -46. 



The value of c- may also be calculated from an equation, to be 

 given hereafter, for determining the angle of elevation y from 

 the angle of the tangent cone, whenever the value of y can be 

 measured directly. 



16. In the case of the conical shells of the Orthocerata, in 

 which we can no longer speak of the angle of retardation, but 

 must measure the apical angle of the cone, the above methods 

 are not applicable. Here, however, the apical angle can 

 always be directly observed on the sides of the shell : or if it 

 have a known number of longitudinal ornaments, by taking 

 that multiple of the angle between two of them and dividing 

 by 27r we obtain the sine of half the required angle if the sec- 

 tion be circular. It may also be obtained for any particular 

 plane, whatever be the section, by the formula 



—^ — = tan^ or siiirj 

 2s 12 



if d 1} d 2 be two diameters in that plane, and s the distance be- 

 tween them measured either on the axis or on the slant side. 

 It would be unnecessary to remark that it is the difference 

 and not the ratio of the diameters which gives us the apical 

 angle, were it not that this ratio is given by some palaeontolo- 

 gists among the shell-characters, where of course it is per- 

 fectly useless. Methods quite analogous may be used for 

 curved shells where the pole is known. For if with pole as 

 centre any circle be described cutting the two curves, the angles 

 between the tangents or between the radii corresponding to 

 the points of intersection will be the /3 required. Or if d l7 d 2 

 are the distances between these points for two circles, 



d 1 —d 2 . /8 , d 1 —d 2 . fi 



-fr-, ; = sin ^-, and — ^ = sm ¥ . cos a, 



2yr^ — r 2 ) 2 ; 2s 2 



r i> r 2> s being taken on either curve (see fig. 8). 



If the transverse section of a conical shell is elliptical, the 

 apical angle will not, of course, be the same in the planes of 

 the major and minor axis, though, the ratio of these being 

 constant, one angle may be deduced from the other. When 

 curvature takes place in the plane of the major axis, the curves 

 formed by the ends of the minor axis do not lie in one plane, 



