FORM OF A SPONGE 319 



or 



\a-nF/ x' p \~ .3/ 3 n 



.-.when ^. = y_^^._ = .^^-. (9) 



(Note that the value of G is not involved in this equation.) 



F P 



.-. x^ = l-OSlp -^ , or taking p = 1-025, x^ = 1-11 -^ • 



The negative and imaginary roots do not concern us, for x 

 is necessarily rational and positive ; and since P and F are 

 finite and positive, this equation gives a finite and rational 

 value for x, and therefore, from (5), I is also rational, positive 

 and finite. 



But there is no jet from an aperture of infinite radius, 

 because the velocity is zero, and there can be no jet from 

 a closed aperture ; 



therefore when x= cc, 1 = 0; and when x = 0, 1 = 0; 



therefore the value of x given by (9) corresponds to a greater 

 value of I than that when .i- = 0, or when x = cc ; and as it is 



the only positive and finite value for x for which — = 0, 



therefore the corresponding value of I is the only positive 

 maximum of I, and the length of the oscular jet has its 

 greatest value when the diameter of the osculum has the 

 value X, where 



Z=^l.llp=l-03^-^. (10) 



If a second sponge precisely similar to A. 11 were to have the oscular 

 end of its cloaca united with the oscular end of the cloaca of A. 11, to 

 make a twin sponge with a single osculum, we should have twice the 

 number of afferent and efferent canals, &c., and two cloacae ; so that in 

 the computation of F, N and « would both be doubled, with the result 

 that, comparing F^ for the twin sponge with F of the original sponge, 



by(l) 



