XI] OF SHELLS GENERALLY 543 



the opposite side of the radius vector to P, and is therefore 

 imaginary. This final condition is exhibited in Argonauta. 



The Umiting values of A are easily ascertained. 



In Fig. 279 we have portions 

 of two successive whorls, whose 

 corresponding points on the same 

 radius vector (as R and R') are, 

 therefore, at a distance apart 

 corresponding to 27r. Let r and 

 >•' refer to the inner, and R, R' to "'" 



the outer sides of the two whorls. Then, if we consider 



it follows that R' = ae(^+2-)cota^ 



and / = Aae(' + 2.)cOta _ ^^(e + 2:r-^)C0ta^ 



Now in the three cases {a, b, c) represented in Fig. 278, it is 

 plain that r' = R, respectively. That is to say, 



and Ae2'^«ot«|L 



The case in which Ae'^" *^'**^ " = 1, or — log A = 27t cot a log e, is 

 the case represented in Fig. 278, b : that is to say/ the particular 

 case, for each value of a, where the consecutive whorls just 

 touch, without interspace or overlap. For such cases, then, we 

 may tabulate the values of A, as follows : 



