534 PROCEEDINGS OP THE ACADEMY OF 



But we have seen (Fig. 3) that // = F A = a -f s/a (a m) and f'^FB 



= a y/ a{a m) ; hence / //' 2 v a ( a m). Further, in our 



m' 2 a v a (a m) 

 former work, we have shown that p = - and e = 2 5 ; 



p m~ a p 



hence = . ^^r. Substituting these values of/)' /' and of 



e va(d m) e 



in equation (3), there results 



V=4 *m~ a (4) 



which is the volume of a prolate ellipsoid whose semi-transverse axis is a and 

 semi-conjugate m. 



The further discussion of the hyper-ellipse has led us to some interesting 

 formulse, which, whether this curve is really important in Morphology or not, 

 appear worthy of attention in a mathematical point of view. These formulae 

 we hope to present in a subsequent paper. 



CONCLUDING REMARKS. 



The coincidence in form between organic outlines and mathematical figures 

 i? a subject of difficult interpretation. It will, therefore, be sufficient for me, 

 without expressing a confident opinion as to the meaning of such resemblances, 

 to say that my study of the subject has induced the belief that all the resem- 

 blances of this kind which have been found are not mere accidental coinci- 

 dences, but that some of them are the results of a mathematical arrangement 

 in nature. The reason or the fitness of this arrangement, I am inclined to 

 think, is explicable mathematically, at least to a greater extent than has been 

 generally supposed, and the search for such an explanation I conceive to be a 

 legitimate effort of science. I would, therefore, encourage the collection of 

 drawings of organic objects and of mathematical figures, together with other 

 materials for study, in order to combine facts as they appear, and prepare 

 for the discussion of definite questions which may be suggested. Works or 

 memoirs devoted to the measurement of organic products, constituents, and 

 combinations, or of organic functions, as measured by phenomena of produc- 

 tion, of motion or duration, or by the evolution or abstraction of force, may 

 probably be useful in furnishing data for the study of the cause of organic 

 forms. But we shall not prescribe rules in this respect. Of late years many 

 researches of this kind have been made, and in several cases by naturalists 

 or physiologists who have not only united mathematical ability to other ac- 

 quirements, but have left the records of their labor in the shape of mathe- 

 matical formula?. Some of these formula? may become useful in studying the 

 cause of organic forms. 



For some time the writer has been engaged in collecting materials of the 

 kind described. The preparation of this paper was undertaken from a desire 

 to render useful the labor incurred in the collection ; but circumstances have 

 prevented as full a treatment of the subject as we could have wished. Many 

 of the authors cited deserve more attention than we have been able here to 

 give to them, and others have been left unnoticed because our limits did not 

 permit us to speak of them as they deserve. On another occasion we hope to 

 return to our subject under more favorable circumstances. 



In submitting this paper to the reader, I have endeavored to make the best 

 selection of matter for general perusal, and to supply some desirable refer- 



[Dec. 



