544 



THE LOGARITHMIC SPIRAL 



CH. 



We see, accordingly, that in plane spirals whose constant angle 

 lies, say, between 65° and 70°, we can only obtain contact between 

 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 constant 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 y%. 



Fig. 280. 



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 separation or of overlap 

 which are exemplified in Fig. 278, a and c. And, just as we 

 have constructed this table of values -of A for the particular case 

 of simple contact between the whorls, so we could construct 

 similar tables for various degrees of separation, or degrees of 

 overlap. 



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

 in which the interspace between the whorls is everywhere a 

 mean proportional between the breadths of the whorls them- 

 selves (Fig. 280). 



