If on the other hand we have, for any given value of α, a value of λ 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 λ 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 themselves (Fig. [280]). {545}
In this case, let us call OA = R, OC = R1 and OB = r. We then have
R1 = OA = a eθ cot α ,
R2 = OC = a e(θ + 2π) cot α ,
R1 R2 = a e2(θ + π) cot α = r2 [†].
And
r2 = (1 ⁄ λ)2 · ε2θ cot α ,
whence, equating,
1 ⁄ λ = eπ cot α .