62 EEPOET— 1881. 



may be easily shown. In Clebscli's form d4'/cU= — {21-^,+ v\pi^+ iv\p.) 

 since c^L/vt = 0. Putting in this and substituting for m i' iv their values, ic 

 reduces to 



^- - V = - ^ -h(<t>/ + V'/+ i>.')+hn'- ( ^.,' + 4v + 4^.') 



which is the same as Hill's form reduces to, with the sign of V changed, 

 when their values are substituted for u v 10. In the latter part of the 

 paper is proved a theorem analogous to Helmholtz' law in vortex motion, 

 as to the constancy of vortex strength x section along a vortex filament.' 

 Expressed in quatei'nion symbols, which enable us to see the meaning 

 much more clearly, if Q,il be the surfaces which by their intersections 

 give the filaments, it is easy to show that 



(area of section of filament) x T V (AQ.AR) = const. 



Combining this with the expression for the vortex -^V (v'W-v4') given 

 above, and limiting Q.R to the case where they give the vortex filaments, 

 Helmholtz' law follows as a case of a more general theorem. A special 

 case of the preceding was worked out by Mr. Craig- towards the end of 

 the same year. The case only of steady motion is considered and that 

 only when of the three surfaces two are chosen to give the vortex lines. 

 As this part of the paper is almost the same as Hill's, modified so as 

 to leave out of consideration the unsteadiness of the motion, there is 

 no need to refer to it further here. The latter part of the same paper 

 will be noticed again under vortex motion. 



A most powerful method of attacking particular problems in fluid 

 motion is that known as the method of images. The conception appears 

 to have been first introduced by Stokes^ in 1843, in considering the 

 problem of the motion of a sphere in presence of a plane, but the theory 

 received no extension until Thomson's discovery of the electrical image 

 of a point of electricity in presence of a conducting sphere again drew 

 Stokes' attention to the matter, the result being a short note in the 

 ' British Association Heport ' for 1847, in which he gives the image in a 

 sphere of a doublet whose axis passes through the centre of the sphere. 

 The general cases for a source of fluid and for a doublet with its axis in 

 any direction have been published by the Avriter,* and for a vortex by 

 Mr. Lewis. ^ The images in two dimensions for a circle have been long 

 known, but I have not been able to discover by whom first used. The 

 images for ellipses are also known.^ These will be referred to again 

 under the head of special problems. 



Maximum and Minimum Theorems. Uniqueness of Solution, Sfc. 



Befoi-e going further, it may be well to refer here to one or two points 

 in the history which are of importance and which have not yet been 

 touched upon. It has been shown how Clebsch deduced a maximum and 

 minimuna theorem on the equations of motion ; but the first, so far as I 

 am aware, to give any such general theorem peculiar to hydrodynamics, 



' It is not stated quite correctly, as he sjDeaks of the prodvict of a vector and an 

 area being constant, instead of the tensor of a vector. 



'^ ' Methods and results. General properties of the eqnations of steady motion,' 

 United States Coast and Gcmlctic Siirrei/. Washington, 1881. 



3 ' On some cases of fluid motion,' Camp. Phil. Trans., viii. 



* ' On the motion of two spheres in a fluid,' Roij. Soc. Trans., pt. ii. 1880. 



'i ' On the images of vortices in a spherical vessel,' Quart. Jour., xvi. 



^ ' On functional images in ellipses,' Quart. Jour., xvii. 



