92 PROC. ENT. SOC. WASH., VOL. 24, NO. 4, APR., 1922 



at least in part to inaccuracies in calculation and to an imper- 

 fect choice of constants in the formulae. After this point, how- 

 ever, as the figures for the length of the radius vector show, the 

 differences between the two curves become more and more 

 marked until the vectorial angle approaches 300 degrees after 

 which the curves again become similar. 



The complete curves are represented in figures 20 A and 20 B 

 and it is obvious from these that the mere inspection of the 

 portions of the curves on the left hand side of the figures enables 

 us to classify them at once as distinct morphological species. 

 The curve A is of course the circle, whose formula, in polar 

 coordinates is in this case 



r 2 2Rr cos e + R 2 a 2 = O 

 the circle plotted being represented by the equation 



r 2 ~2.7r cose 5.2 = O 

 while the curve B is the one known as "Pascal's snail," whose 



formula is 



r = a cos e a 



where r = length of radius vector, R = length of radius of the circle 

 plotted, a = a constant and e=the vectorial angle: the value of 

 a in the equation for the curve B being in this case, 2. 



Figure 20. A, circle, B, "Pascal's snail" or cardioide. 



Thus, from two forms which are at least in practice morpholo- 

 gically indistinguishable one from another but nevertheless 

 develop according to different laws of growth, there may arise 

 forms morphologically quite different: or conversely, two 

 forms which are at a certain stage of development morphologi- 

 cally very dissimilar may in later stages become so much alike 

 that it is practically impossible to refer them to different species: 

 and this is precisely the phenomenon which we have found to 

 exist in the case of the species studied in this paper, or more 

 generally, the phenomenon of pcecilogony. 



