Curves formed by Cephalopoda and other Mollusks. 261 



and by direct measurement we find that 



E(e,<£) + E(e, </>,) = 1-037; 

 whence, from (6), 



«*(«'-•)«*«_ 1-139 



(the coefficient on the right-hand side becoming 1 + a / ). The 

 second side of (1) becomes now 1*184. This value does not 

 agree sufficiently with 1*071 for the difference to be due to 

 any errors of observation ; and it follows that" Ammonites pla- 

 norbis was produced from a more involute shell than that called 

 Ammonites erugatus : and such more involute varieties occur 

 in the south of England. 



19. To find the angle of elevation in a turbinated shell. 



The point in the tracing-curve whose angle of elevation is 

 measured must, of course, depend on the shape of that curve. 

 When it is such as to form a complete cone, as in JEulima, the 

 angle for any point on the surface is directly measurable. 

 When the curve is such as the ellipse, the angle measured by 

 the tangent lines does not correspond to the 7 already used, 

 but is connected with it by an easily expressed relation ; for 

 if co be the semi- vertical angle of the tangent cone, the equa- 

 tion (A) must give equal roots when r cot co is written for z, 

 which requires that 



tan y = cot &) + cr \/ /c 2 (cot co cot e + l) 2 + (cot co — cot e) 2 . 



When, as is often the case, co = e, this reduces to 



COS CO + a/C 



tan 7 = 



sin ft) 



It is obvious that when &) and y can both be observed directly, 

 this gives an equation to find <r, as noted in § 15. 



An example of the use of this equation may be taken from 

 the previously quoted Cyclostoma elegans. Here 



<r=-46, ft) = 24°, € = 156°, *=-87315, 

 whence 



tan 7 = 4-874 = tan 78° 24 ; , 



the angle derived from direct measurement being 78^°. 



20. The semi- vertical angle of the cone for any particular 

 point may also be calculated by means of other angles often 

 more easily measured. Let co be such an angle, and i the angle 

 between the tangent at that point and the axis. Let AB (Sa) 

 be an element of the curve at that point, DB its projection on 

 the plane of xy (8s). Then 



dz = da cos i, ds = da sin i, dz = dr cot co, dr = ds cos « ; 



