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



of 29 millims., being a vertical ellipse. In this case the um- 

 bilicus, not being approximately in one plane, has to be cal- 

 culated : we obtain R= 1*378, whence 



X==1 ~56( 1+ F378)> 



Q log A, 1-673713 0Q _ 



P = 7T — r^-p- =7r—— ™- = 29-66 tt; 



or the inner edge of the shell represents the surface nearly 



fifteen whorls behind. 



The other two formulae may be illustrated on Ammonites 



lawiusculus. The measurements along one diameter of 50 



millims. were — outer whorl 21^, umbilicus 14, remainder 14§ ; 



whence 11 = 1-47 and X = *28. Substituting these values, 



,. . 1--6050 3950 _ v , . 



we obtain x— ^ tt^ = coo. = 67. Jb rom actual measure- 



1 — -4116 5884 



ment the whorls are \\ concealed, i. e. '66. The value of a 

 corresponding to It =1*47 is 83°. The breadths of the whorls 

 at two points distant 67 millims. of arc are 14J and 20 mil- 

 lims. respectively; whence 



5-6667 = 67 x -1218 (1-X), 



which gives \=*32. This latter measurement is the least 

 trustworthy, especially if the direction of the radii cannot be 

 accurately ascertained. 



15. In turbinated shells whose elliptic axis is oblique, we 

 require to know the retardation of the inner extremity of the 

 major axis, i. e. the value of cr. This is best done by compa- 

 ring the diameter perpendicular to the axis with the major 

 axis of the last whorl. To determine this diameter we must 

 add the two radii obtained from the condition that z shall have 

 equal roots in the equation A which correspond to 6 and 6—ir. 

 This condition gives us 



r = e ecotx (l±a*/ k 2 cot 2 e + 1). 



The smaller of these values corresponds to the radius, if any, of 

 the open umbilicus, as in § 5 ; the larger is the one we require. 

 Therefore, if d be the measured diameter, 



d = e 9cota (l + e- ncota ) \l + a */k 2 cot 2 e + l\ ; 



d<7Cosece = a{l + *r ffCOta }{l + o-\/*: 2 cot 2 e + l}. 

 For example, in the Cyclostoma elegans before mentioned, 



