Mass carried forward by a Vortex. 603 



Nowa/3=2-0S72 V 3 = 3-6151, 



E 2 = f\<log>--2740-/ 2 k 



. E 3 = /xUa-ix 3-6151 U 2 



= ix4'38497ra 2 U 2 



= •1339^. 



7T 



The last result can be verified from the fact, that since the 

 form of the translated mass differs only inappreciably from 

 that of an elliptic cylinder, the energy of the external motion 

 is aa//3a of the liquid displaced by the cylinder. Now 

 a//3 = l'205s The difference from 1*2130 is probably due 

 more to the Telocity changes due to the slight change in 

 form, than to the difference in area of the two cross sections. 



The curious fact emerges, that so long as the filaments are 

 not so close as to appreciably affect the shape of their cross 

 sections, the energy of the external fluid is constant and is 

 independent of the velocity of propagation. As the velocity 

 increases, the quantity carried forward diminishes so that 

 this result follows. This fact might have been foreseen from 

 a consideration of dimensions, remembering that with the 

 proviso above, there is only one length a at disposal to define 

 the system. 



The most interesting portion of the motion, when the two 

 filaments close in to be almost in contact, is unfortunately 

 not at present capable of being treated. It is easy to show, 

 however, that if the shape of the core-section referred to its 

 centre of gravity be given by r = a(l -f /> 2 f 2 cos 20+ . . .) 

 where f=c/a, the change in U depends on £ 4 . The above 

 values are therefore approximately good even when a con- 

 siderable amount of deformation is present. 



(b) 7 lie circular ring. 



When with diminishing energy the aperture just closes 

 up, the aggregate, as is well known, takes the form of a 

 sphere. 



