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 × 4 × 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 α + β, while the pattern of the developed cercaria is expressed by 4 (α)2 + 4 (β)2.
4. The fundamental pattern of tie cercaria is likewise found in the sporocyst.
5. These data have a significant bearing on the application of mathematics to the problems of growth and development.