XI] 



ITS MATHEMATICAL PROPERTIES 



533 



a radial direction of any two whorls {W). We have then merely 

 to apply the formula 



9 cot a- 



which we may simply write r = e^^^^"^, etc. ; since our first radius 

 or whorl is regarded, for the purpose of comparison, as being equal 

 to unity. 



Thus, in the diagram, OCIOE, or EF/BD, or DC/EF, being 

 in each case radii, or diameters, at right angles to one another, 



are all equal to e^^° ". While in like manner, EO/OF, EG/FH, 

 or GO/HO, all equal e'^^°*^ and BC/BA, or CO/OB = e^''''''^\ 



Fig. 270. 



As soon, then, as we have prepared tables for these values, 

 the determination of the constant angle a in a particular shell 

 becomes a very sirhple matter. 



A complete table would be cumbrous, and it will be sufficient 

 to deal with the simple case of the ratio between the breadths of 

 adjacent, or immediately succeeding, whorls. 



Here we have r = e27rcota^ ^j. j^^g ,^. = log e x 27r x cot a, from 

 which we obtain the following figures * : 



* It is obvious that the ratios of opposite whorls, or of radii 180° apart, are 

 represented by the square roots of these values ; and the ratios of whorls or radii 

 90° apart, by the squaie roots of these again. 



