256 Rev. J. F. Blake on the Measurement of the 



but on a cone whose vertical angle 2Sis given by cot 8 = k^. — -, 



1 — A, 



the major axis being supposed to be perpendicular to the axis 

 of the shell. This, of course, is not the tangent cone to the 

 whorls. It follows that if the curves in the plane of the major 

 axis remain regular, and yet one plane can be made to touch 

 the surface at the extremities of all the minor axes, the shell 

 must either be unsymmetrical or the shape of the section must 

 vary ; and the latter will certainly be the case when tivo such 

 planes can be found. We cannot, therefore, measure the apical 

 angle of a Cyrtoceras or Dentalium, in a plane perpendicular 

 to that of the curvature, with perfect accuracy. 



17. When a discoid shell has suffered distortion (that is, 

 compression in the plane of the major axes), /3 may still be 

 found if more than half a whorl be left ; but no method suffi- 

 ciently simple to be of use can give either a or fi on distorted 

 fragments. A more important case is when the compression 

 is perpendicular to the plane of the major axis. This has no 

 effect on a, but increases /3 at the expense of tc. In measuring 

 the apical angle of Orthocerata, we therefore require the fol- 

 lowing : — 



Given the semi- vertical angle of a cone of elliptical section 

 ivhen the excentricity of the section is e, to find it ivhen the 

 same is compressed so that the excentricity is e'. 



Describe a sphere with the vertex as centre ; then the cone 

 will cut this sphere in a curve of double curvature, the length 

 of which will remain constant when the excentricity is altered, 

 since the area of such a cone w r ill be unaltered by compression. 



If 7T— 2w be the vertical angle of the cone in the plane of 



the major axis 2 a, and h the height, then - = tan co. 



Let <f> be the complement of the excentric angle at any point 

 of the ellipse, whose excentricity is e, and p the radius from the 

 centre of the section 8, da the element of the arc on the sphere, 

 ds of the ellipse, dd the angle between p and p + 8p } A the radius 

 of the sphere ; then 



dcr = Ad6, 



p\W 2 = ds 2 -dp 2 , 

 and p~ \/A 2 + a 2 (l — e 2 cos 2 <£), 



and ds = a\/l — e 2 sin 2 (f>d(f>; 



r1 n <Sh 2 + a 2 (l-e 2 )-h 2 e 2 sm 2 cf> .. 

 d6=a W(l-ecos^) d *> 



^-A ^ se ° 2 ^ ~ e2 ( ! + tan " °> sin ' 2 <ft ) d<t>, 

 sec 2 o) — e 2 cos 2 </> 



