528 



PROCEEDINGS OF THE ACADEMY OP 



The curve can also be constructed by drawing F P from any assumed pole 

 at F, at any angle with an assumed axis A B. Then, knowing the length and 

 thickness of the egg and the angle AFP, the length F P can be calculated by 

 the aid of a proper formula, hereafter given, and transferred to the drawing. 



COMPARISON OF EGGS WITH THE HYPER-ELLIPSOID. 



Egg of Planorbis corneus. The example just given to show the construc- 

 tion of the hyper-ellipse (Fig. 3) presents a good imitation of the magnified 

 drawing of an egg of Planorbis corneus.* It is remarkable that the focus F 

 of the theoretical egg falls, as nearly as can be readily observed, in the centre 

 of the vitellus, according to the engraving given by Jacquemin. The magni- 

 fied egg of the engraving measures, say, length 1*63 inches, thickness 1*31. 

 The distance B F is found by calculation -45+. 



Egg of Epiomis. The cast of the egg of Epiornis belonging to the Acade- 

 my of Natural Sciences in this city is, I doubt not, from the pointed egg de- 

 scribed by M. St. Hilaire.f A longitudinal section of this cast was obtained 

 by cutting a templet to fit closely around it, then tracing the form of the egg 

 from the templet. The first section thus obtained was not quite symmetrical 

 with respect to the long axis. A second section, taken on a plane at right 

 angles to the plane of the first, was more nearly symmetrical. The cross-sec- 

 tion of the cast measured so nearly circular, that the small difference in the 

 diameter of the sections taken may be disregarded. For the purposes of calcu- 

 lation we have taken M. St. Hilaire ? s dimensions reduced to inches, length 

 12-756, thickness 8-859. 



Fig. 4 represents the theoretical egg on a scale of one-fourth. The follow- 

 ing tables exhibit the dimensions of the two real sections and of the theoretical 

 section. Each real section is divided by the axis into two parts not entirely 

 symmetrical. The radii vectores of each part are given for various polar 

 angles. The distance from the obtuse end of the egg to the line of greatest 

 thickness is not involved in the construction of the theoretical ovoid. This 

 distance as measured on the egg should be compared with the same as found 

 by construction. In the tables it is designated by B' -f- p cos y. 



TABLE I. 



MEASURED VALUES FOR REAL EGG. 



Note. The distance from the large end of the egg to the greatest cross- 

 section = B' + P cos > = 6 inches. The radius for a polar angle of 56 34' 

 is 5-39, average of four dimensions. 



* Memoire sur l'Histoire du Planorbis corneus, par Emile Jacquemin. Nova Acta Acad. C. 

 Leopold, vol. xviii. p. 638. 



f The agreement in size is satisfactory, and the egg is marked with the name of Dr. Warren, 

 who relates that a cast of the pointed egg was presented to him. (Fossil Impressions, &c, Boston, 

 19C>4.) The length which he gives for the egg is incorrect. The mistake probably arose from a 

 typographical error, which is corrected in Comptea Rendus, vol. xl. p. 519. 



[Dec. 



