E. C. Faust 
211 
twice, for it is represented not only in the main tubules but also in each of the 
tertiary tubule afferents. Beyond this is the four-fold division into capillaries 
which apparently represents an addition to the fundamental formula. The 
more complete formula then is (a) 2 + (£) 2 , while 4 (a) 2 + 4 (j3) 2 represents the 
flame-cell total for each side of the cercaria. 
Discussion. 
The problem presented in this study is a fundamental one. The experi¬ 
mental biologist has shown that the organism obeys within limits certain 
definite laws, some of which, such as the cleavage of the embryo and the 
division of sex cells in the earlier stages, are mathematically precise and pre¬ 
determined. The problem of the excretory system in the trematode is no less 
a problem in mathematical development. I have previously shown how this 
obtains in the amphistome group (Faust 1919), in the echinostomes, xiphidio- 
cercariae and furcocercariae of the distome group (Faust 1919 b). Cort has 
confirmed this in his several studies on distome species. Moreover, these 
larval studies are supported by the monumental work of Looss (1894, 1896) 
on adult forms. In unpublished studies of holostomes and aspidobothrids 
I have found the same definite plan of development. 
It is of equal significance that the fundamental pattern of excretory tubules 
of a particular species (representing a group) obtains for the parthenita (sporo- 
cyst or redia) of that species. This has previously been demonstrated in 
Cercariaeum mutabile (Cort 1919), in the amphistome, Cercaria convoluta 
(Faust 1919) and in the monostome, Cercaria spatula (Faust 1919 b). The 
present paper shows it to obtain in Cercaria pekinensis. The pattern is ap¬ 
parently so potentially inherent in the protoplasm that, through the several 
generations, even profound alterations in other systems and tissues and, 
indeed, in habits of living, including temperature, tonicity, and H-ion con¬ 
centration, have practically no effect on this system. 
Summary. 
1. Cercaria pekinensis , a new cystocercous cercaria, is described. 
2. The excretory system of the developed cercaria consists of a bladder, 
a drainage channel through the tail and a pair of anterior collecting tubules, 
each of which receives eight main (tertiary) branches. Each branch is composed 
of 2 x 4 x 4 units. The posterior-most branch cares for the drainage of the 
tail, including the corresponding flapper. 
3. A study of representative stages in development shows that the common 
denominator of the excretory system is a + yS, while the pattern of the de¬ 
veloped cerqaria is expressed by 4 (a) 2 + 4 (/2) 2 . 
4. The fundamental pattern of the cercaria is likewise found in the sporo- 
cyst. 
5. These data have a significant bearing on the application of mathematics 
to the problems of growth and development. 
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