i 7 o HERMANN VON HELMHOLTZ 



theorems and conclusions, taken as pure mathematics, are 

 fundamental laws of the modern Theory of Functions. His 

 conclusions for ring-shaped vortex -filaments are very inte- 

 resting. When two such rings of small section have the same 

 axis and the same direction of rotation, they travel in the same 

 direction; the first ring widens and travels more slowly, the 

 second shrinks and travels faster, till finally, if their velocities 

 are not too different, it overtakes the first and penetrates it, 

 the same process is then repeated for the next ring, and so 

 on. An analogous statement holds good when the directions 

 of rotation are opposite. 



When Tait, some ten years later, proposed to translate 

 the work of Helmholtz on Vortex- Motion, and wrote to him 

 on the subject, Helmholtz replied : ' If you find quaternions 

 useful in this connexion, it would be highly desirable to draw 

 up a brief introductory explanation of them, so far as is 

 necessary in order to make their application to vortex-motion 

 intelligible. Up to the present time I have found no mathe- 

 matician, in Germany at any rate, who was able to state what 

 quaternions are, and personally I must confess that I have 

 always been too lazy to form a connected idea of them from 

 Hamilton's innumerable little notes on the subject." 



In 1868 Bertrand published some criticisms in regard to the 

 universality of Helmholtz's methods. Helmholtz had assumed 

 (as above) that the motion of an indefinitely small volume of 

 water is due to the propagation of an element of fluid through 

 a space, the expansion or contraction of the element in three 

 principal directions of dilatation (so that a rectangular parallele- 

 piped constructed of water, whose sides are parallel with the 

 principal directions of dilatation, remains rectangular, while 

 its sides alter in length, but remain parallel with their previous 

 direction), and to revolution round a definite instantaneous 

 axis of rotation. Bertrand contended that in a great number 

 of cases oblique parallelepipeds might also be constructed 

 with an arbitrary direction of the edges, which could be 

 transformed into other parallelepipeds, whose edges should 

 remain parallel with those of the former. Helmholtz replied 

 in three Notes published in the Comptes Rendus for 1868, 

 'Sur le mouvement le plus general d'un fluide/ 'Sur le 

 mouvement des fluides,' and 'Reponse a la note de M. J. 



