WOLFFIAN TUBULES IN MAMMALIAN EMBRYOS 427 



principal references to the double spiral which follows the simple 

 S and which is here regarded as the common origin of the later 

 patterns and the key to their interpretation. 



A suitable designation for the form which we have called a double 

 spiral (fig. 3) is unfortunately not available. It is a shape which occurs 

 at the apex of the coil of large intestine in the pig, where some of its 

 interesting properties have been discussed by Lineback, and it may 

 perhaps be found elsewhere. Impressed with Thompson's application 

 of mathematics to biological problems, the aid of a mathematician was 

 sought in describing the form in figures 3 and 4, and in determining 

 what factors of growth might account for the regular transformation 

 of the simple S into the double spiral. There are, however, so many 

 unknown factors in the nature of the surrounding medium, which may 

 keep the growing coil within a circular area, and in the physical nature 

 and method of expansion of the tube-wall itself, that a mathematical 

 interpretation was not forthcoming, and the problem was returned as 

 'biological.' It may be noted that mitotic figures are rather evenly 

 distributed, and that no evidence is found to support the idea of Mihal- 

 kovics that a special elongation of the middle segment in cramped 

 quarters produces the double spiral. In fact, if the middle portion 

 remains short and the two ends of the simple S elongate, continuing 

 their respective curves, the double spiral will result, though the axis 

 of the figure rotates. In the shifting position of the Wolffian duct in 

 later stages, there is clear evidence that such a curved elongation of the 

 distal segment actually takes place. 



In human embryos the simple S may be expected in 6-mm. 

 specimens and in those somewhat smaller. At 7.5 mm. the 

 double spiral has formed, as shown in the model, figure 5. The 

 tubule chosen was one in which the essential bends fell within 

 the plane of section, thus lessening the chance of distortion in 

 the process of modeling. The liberty has been taken of showing 

 the tubule reversed in the figure, as if it belonged to the left 

 Wolffian body, to facilitate comparison with other models. Ex- 

 Fig. 5 Embryo of 7.5 mm. Harvard Embryological Collection, series 256, 

 sections 471-485. 



Fig. 6 16.0 mm. Harvard Embryological Collection, series 2044, sections 

 1045-1067. 



Fig. 7 22.8 mm. Harvard Embryological Collection, series 871, sections 

 887-908. 



Fig. 8 Diagram of the double-spiral stage, shown in figure 5, with its sub- 

 division into segments C, TJ, and Z. 



Fig. 9 A corresponding diagram of the stage shown in figure 7. 



