XI] 



OF SHELLS GENERALLY 



801 



consecutive whorls if the rate of growth of the inner border of the 

 tube be a small fraction — a tenth or a twentieth — of that of the 

 outer border. In spirals whose con- 

 stant angle is 80°, contact is attained 

 when the respective rates of growth 

 are, approximately, as 3 to 1; while 

 in spirals of constant angle from about 

 85° to 89°,contact is attained when the 

 rates of growth are in the ratio of from 

 about f to j%. 



If on the other hand we have, for 

 any given value of a, a value of A 

 greater or less than the value given 

 in the above table, then we have, 

 respectively, the conditions of separa- 

 tion or of overlap which are exempUfied 

 in Fig. 385, a and c. And, just as we 

 have constructed this table for the 

 particular case of simple contact, so we could construct similar tables 

 for various degrees of separation or of overlap. 



For instance, a case which admits of simple solution is that in 

 which the interspace between the whorls is everywhere a mean pro- 

 portional between the breadths of the whorls themselves (Fig. 387). 

 In this case, let us call OA = R, OC = R^, and OB = r. We then 

 have 



Fig. 387. 



i^2=OC = ae(^+2'^>^^*^ 



And 



r2= (1/A)2 . €2^«0t«, 



whence, equating. 



1/A = e 



TTCota 



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

 that, because the interspace AB i& & 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.. 



