QUATERNIONS AND VECTOR ANALYSIS. 175 



which I have used by quoting a sentence addressed to the British 

 Association a few years ago. The speaker was Lord Kelvin. 



" Helmholtz first solved the problem Given the spin in any case 

 of liquid motion, to find the motion. His solution consists in finding 

 the potentials of three ideal distributions of gravitational matter 

 having densities respectively equal to I/TT of the rectangular com- 

 ponents of the given spin; and, regarding for a moment these 

 potentials as rectangular components of velocity in a case of liquid 

 motion, taking the spin in this motion as the velocity in the required 

 motion " (Nature, vol. xxxviii, p. 569). 



In the terms and notations of my pamphlet the problem and 

 solution may be thus expressed : 



Given the curl in any case of liquid motion to find the motion. 



The required velocity is l/4?r of the curl of the potential of the 

 given curl. 



Or, more briefly The required velocity is j- of the Laplacian 

 of the given curl. 



Or in purely analytical form Required w in terms of Vx, when 

 V.o> = 0. 



Solution : w= !/47rVxPot Vx<o = 1/4-Tr Lap Vxo>. 



(The Laplacian expresses the result of an operation like that by 

 which magnetic force is calculated from electric currents distributed 

 in space. This corresponds to the second form in which Helmholtz 

 expressed his result.) 



To show the incredible rashness of my critics, I will remark that 

 these equations are among those of which it is said in the original 

 paper (Proc. RS.E., Session 1892-93, p. 225), " Gibbs gives a good 

 many equations theorems I suppose they ape at being." I may add 

 that others of the equations thus characterized are associated with 

 names not less distinguished than that of Helmholtz. But that to 

 which I wish especially to call attention is that the terms and 

 notations in question express exactly the notions which physicists 

 want to use. 



But we are told (Nature, vol. xlvii, p. 287) that these integrating 

 operators (Pot, Lap) are best expressed as inverse functions of V. 

 To see how utterly inadequate the Nabla would have been to express 

 the idea, we have only to imagine the exclamation points which the 

 members of the British Association would have looked at each other 

 if the distinguished speaker had said : 



Helmholtz first solved the problem Given the Nabla of the velocity 

 in any case of liquid motion, to find the velocity. His solution was 

 that the velocity was the Nabla of the inverse square of Nabla of 

 the Nabla of the velocity. Or, that the velocity was the inverse 

 Nabla of the Nabla of the velocity. 



