800 THE EQUIANGULAR SPIRAL [ch. 



externally from (4), where P' lies on the opposite side of the radius 

 vector to P, and is therefore imaginary. This final condition is 

 exhibited in Argonauta. 



The Hmiting values of A are easily 

 ascertained. 



In Fig. 386 we have portions of 

 two successive whorls, whose corre- 

 sponding points on the same radius 

 vector (as R and R) are, therefore, 

 ^^* ' at a distance apart corresponding to 



277-. Let r and r' 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;r) cot a^ 



and r' = Aae^^+^^^^^o*^ = j^g(^f27r-7)cota_ 



Now in the three cases (a, b, c) represented in Fig. 385, it is 

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



Xae^0+2n)COta j ae^cota^ 



and Ae^^^ota | i^ 



The case in which Ae^'^^^*'^ = 1, or — log A = 277- cot a log e, is 

 the case represented in Fig. 385, 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: 



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

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



