XI] OF SHELLS GENERALLY 545 



lu this case, let us call OA = R, OC = Ri, and OB = t. 

 We then have 



R, = OA = ae^'P^t'^^ 



And r2=(l/A)2.e'^'^°*% 



whence, equating, 1/A = e"*^°*". 



The corresponding values of A are as follows : 



Ratio (A) of rates of growth of outer and .inner 



border, sucli as to produce a spiral witli interspaces 



between the whorls, the breadth of which 



interspaces is a mean proportional between the 



Constant angle (a) breadths of the whorls themselves 



90° 1-00 (imaginary) 



89 -95 



88 -89 



87 -85 



86 -81 



85 '76 



80 -57 



76 . -43 



70 -32 



65 -23 



60 . -18 



55 . -13 



50 -090 



45 063 



40 -042 



35 026 



30 -016 



As regards the angle of retardation, j8, in the formula 

 r' = Ae'*^°t% or / = e(»-«cofca^ 

 and in the case 



/ = e<2.-^)cota^ Qj. - log A = (277 - ^) cot a, 



* It has been pointed out to me that it does not follow at once and obviously 

 that, because the interspace AB is a mean proportional between the breadths of 

 the adjacent whorls, therefore the whole distance OB is a mean proportional 

 between OA and OC. This is a corollary which requires to be proved; but the 

 proof is easy. 



T. G. * ' 35 



