PROFESSOR AT BONN 167 



writes that 'after forty years the impression made on me by 

 that noble head, with its deep clear gaze, its classic and 

 dignified expression, is indelible. Helmholtz was generally 

 cheerful and sympathetic, even playful, and delighted of an 

 evening in reading plays aloud ; he- preferred a character-part 

 in Shakespeare, or some other classic. It was an intimate 

 little circle in which the Helmholtz couple took the lead. 

 Often enough Helmholtz would sit still, plunged in his own 

 thoughts, but I never saw him out of temper, or anything but 

 cordial '. ^ 



The last year of Helmholtz's stay at Bonn was marked by 

 a series of important publications. The complexities of 

 acoustics had induced him two years previously to occupy 

 himself with the application of Green's Theorems to hydro- 

 dynamic and aerodynamic problems. In 1857, m a work of 

 genius that proved him to be a mathematician of first rank, 

 4 On the Integrals of the Hydrodynamic Equations which 

 express Vortex-motion ' (published in Crelle's Journal f. reine 

 u. angew. Mathematik), he gave the solution of some ex- 

 tremely difficult hydrodynamical problems. He rejected the 

 earlier hypotheses, and followed up the analogies between the 

 motion of fluids and the electromagnetic action of electrical 

 currents, which were of so much importance for his subsequent 

 work on the Theory of Electricity and Magnetism. Up to that 

 time the integrals of hydrodynamic equations had been deter- 

 mined almost exclusively on the assumption that the rectangular 

 components of the velocity of each element of the fluid are 

 the differential co-efficients, with reference to the co-ordinates, 

 of a certain function, which Helmholtz termed the velocity- 

 potential an assumption which was lawful so long as the 

 motion of the fluid resulted from the action of forces which 

 had a potential of their own. Helmholtz abolished this limi- 

 tation, and took into account the friction between the elements 

 of the fluid, and against fixed bodies, the effect of which on 

 fluids had not till then been defined mathematically, and 

 endeavoured to determine the forms of the motion which 

 friction produces in fluids. Starting with the equations of 

 motion for the interior particles of a liquid, he pictures the 

 changes undergone by an indefinitely small volume of the 

 fluid in an indefinite fraction of time as composed of three 



