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



shells, in which the angles of the different species do not lie 

 so close together ; and the following inverse Table will be suf- 

 ficient to check the direct measurement of the angle by the 

 observation of ratios : — 



Table of Solutions of R=e 7rCOta . 



d. 



K. 



a. 



E. 



a. 



R. 



o 



89 



106 



o 

 69 



3-25 



o 



49 



11-8 



88 



112 



68 



3-44 



48 



12-7 



87 



118 



67 



366 



47 



137 



86 



1-25 



66 



3-88 



46 



14-8 



85 



132 



65 



412 



45 



16-0 



84 



1-39 



64 



4-38 



44 



173 



83 



1-47 



63 



466 



43 



18-8 



82 



1-56 



62 



4-95 



42 



20-4 



81 



1 65 



61 



527 



41 



22- 1 



80 



1-74 



60 



5-62 



40 



240 



79 



1 84 



59 



5-99 



39 



263 



78 



1-95 



58 



6-39 



38 



28-7 



77 



205 



57 



6-82 



37 



31-5 



76 



216 



56 



7-28 



36 



34-6 



75 



2-29 



55 



778 



35 



380 



74 



2-43 



54 



8-32 



34 



420 



73 



2-58 



53 



8-92 



33 



46-4 



72 



2-73 



52 



9-55 



32 



51 4 



71 



2-89 



51 



10-3 



31 



573 



70 



3-06 



50 



110 



30 



640 



11. These methods suffice for recent or complete shells ; but 

 sometimes a fragment of a single whorl is all that is presented 

 to observation, in which case there are two or three possible 

 methods. 



1st. Suppose the inner as well as the outer edge of the 

 whorls preserved, as A B, C I) (fig. 4). Then, if we draw any two 

 parallel tangents, as E G-, F H, the straight line joining the 

 points of contact must pass through the pole. Hence G E F 

 is the angle required. And even if neither edge be preserved, 

 any pair of longitudinal striae or other ornaments will suffice 

 for the purpose. 



2nd. Let the directions and lengths of a pair of transverse 

 ornaments, which we may assume to make the same angle 

 with the radius, be available (fig. 5) as A C, B D. Then the 

 angle between A C, B D equals the angle between the radii, 



BD 



= say 



we have therefore -r-p = e $cota , which gives a. in 



terms of known quantities. 



3rd. Let the outer edge alone be available. Here we have 

 the simple problem, Given an arc of an equiangular spiral, to 

 find its pole and spiral angle. This admits of two easy solu- 

 tions: — 



