NATURAL SCIENCES OF PHILADELPHIA. 533 



angle has the proper proportion to that of the ellipse, in order to derive the 

 number of arms or rays desired.* 



Cubature of the Hyper-ellipsoid. Let F, Fig. 10, be the pole, P M an infini- 

 tesimal arc, and P F M an elementary triangle of any plane curve, referred 

 to the axis F N, which is also the axis of revolution for the solid. 



The centre of gravity, G, of the elementary triangle P F M, is on D G, 

 drawn parallel to the side P M, and so situated that F D is two-thirds of the 

 radius FP, = | p . When the side P M vanishes, F P will coincide with F M, 

 and the distance from F to G will then equal F D = | p, and the angles M F N, 

 G F N, P F N will all be equal, and each = 6. The distance G N will be 

 F D sin 6 = ? p sin 8 ; and the distance described by G during a revolution 

 of the elementary triangle P F M about the axis F N will be F D 2 a- = J v p 

 sin 8. The area of the elementary triangle is, however, J p- c?8, and the 

 solidity of the conical sheet generated by a revolution of P F M, which is the 

 differential of the solid of revolution, will be, by Guldin's Formula, 



dV = i H p sin 6. J p 2 rf6 = | a- p 3 sin 8 dS (1) 



In the present case this becomes 



lie 



1 



pi p~ e sin 8 \ , 



!- 3 <*> = 



e V 2(l c cos AW 



!(1 ccosfl)-' 



dB is the differential of the radius vector p ; 

 2(1 e cos 8)2 

 so that we have, by substitution, for the solidity of the whole hyper-ellipsoid, 



X8 = Tt ft 

 -i^-^dp (2) 

 . =0 e 

 If the radius for 8 = be denoted by p' and the radius for 8 = tt by />", this 

 equation gives 



V=| 7T P-(p'- p ") (3) 



e 



* We may here call attention to the fact that the radius vector of the hyper-ellipse, for the 

 extremity of the greatest ordinate, is p ya in, that is, this radius is a meau proportional between 

 the half-length and half-width of the figure. This is interesting because Dr. Zeising adopts the 

 mean proportion as a general morphological law; but this proportion of itself cauuot 1)3 satisfac- 

 tory : we require some rule for knowing what objects or parts of objects are to be thus compared. 

 As long as no such rule exists, the comparisons may often seem arbitrary. Dr. Zeising proposes, 

 for the egg-curve, to divide the length into two parts, say a' the greater and m' the lesser; then 

 m' will also represent the half-thickness, and we shall have the proportion a' -f- m' : a' : : a' : m,', 



whence a' = y m! (a' + m'). It is not, however, shown by him that this mean proportion is neces- 

 sarily more significant in Morphology than p ya in above mentioned (Xeue Lohre, p. 228). 

 Dr. Zeising's application (Neue Lehre, p. 301) of the extreme and mean ratio, or golden section, to 

 the division of the circle in phyllotaxis, has received a remarkable confirmation as a law of nature, 

 oy the labors of Hanstein and Wright, before cited. His application of this ratio to the relations 

 of the planetary system seem to me worthy of close study ; but proof is required of a similar 

 significance of this ratio in astronomy and in botany, before we can assume that there is an entire 

 identity between the laws which regulate both the planetary and the phyllotaetic systems. 

 (Neue Lehre, p. 327. Normalverhaltniss, &c, Leipzig, 1856, pp. 2, 45.) 



1862.] 



