90 



SCIENCE. 



[N. S. Vol. XI. No. 264. 



orbits without rotation or whether they 

 undergo rotation along with their motion of 

 translation. This was a critical question, 

 for the failure to press and to answer it 

 seems to have retarded progress for nearly 

 half a century. Lagrange, and after him 

 Cauchy and Poisson, knew that under cer- 

 tain conditions * the differential equations 

 of motion are integrable, but they do not 

 appear to have understood the physical 

 meaning of these conditions. It remained 

 for Sir George Gabriel Stokes to show that 

 the Lagrangian conditions of integrability 

 correspond to the case of no molecular rota- 

 tion, thus clearly distinguishing the two 

 characteristic types of what we now call ir- 

 rotational and rotational motion.f Such 

 was the great step made by Stokes in 1845 ; 

 and it furnishes a forcible illustration of the 

 importance, in applied mathematics, of at- 

 tending to the phj'sical meaning of every 

 symbol and every combination of symbols. 

 Thirteen years later came the remarkable 

 memoir of Helmholtz (1821-1894) on the 

 integrals of the equations of hydrokinetics 

 for the case of rotational, or vortex, motion. J 

 This memoir is alike wonderful for the di- 

 rectness with which the mathematical argu- 

 ment proceeds to its conclusions and for the 

 clearness of insight it affords of the physical 

 phenomena discussed. In short, it opened 



* That is, wlien udx -\- vdy + wdz is a perfect dif- 

 ferential, u, V, w being velocity components ; or, when 

 3i« 3i! 3m 3!(j 3i) 3« 

 3)/ 3z' 32 3j;' 3.(,- 3a:' 

 which are the doubles of the components of molecular 

 rotation, vanish, these latter being the conditions for 

 the existence of a velocity potential. 



tThis disoovei-y of Stokes was announced in his 

 fundamental paper ' On the theories of internal 

 friction of fluids in motion, and of the equilibrium 

 and motion of elastic solids, ' TransacHons of the Cam- 

 iriihje Philosophical Society, Vol. VIII. Reprinted 

 also in his Mathematical and Physical Papers, Vol. I 



J " Ueber Integrale der hydrodynamischen Gleich- 

 ungen, welche den Wirbelbewegungen entsprechen," 

 Crelle's Journal fiir die reine und angewandte Maihe- 

 matik, 1858. 



up a new realm and supplied the results' 

 concepts, and methods which led the way to 

 the important advances in the science made 

 during the past three decades. 



Another powerful impulse was given to 

 hydrokinetics, and to all other branches of 

 mathematical physics as well, by Kelvin 

 and Tait's ISTatural Philosophy — the Prin- 

 cipia of the nineteenth century — the fii'st 

 edition of which appeared in 1867. From 

 this great work have sprung most of the 

 ideas and methods appertaining to the 

 theory of motion of solids in fluids, a theory 

 which has yielded many interesting and 

 surprising results through the researches 

 of Kirchhoff, Clebsch, Bjerknes, Green- 

 hill, Lamb and others. Of prime im- 

 portance also are the numerous contribu- 

 tions of Lord Kelvin to other branches 

 of hydrokinetics, and particularly to the 

 theory of rotational motion.* In fact, 

 every department of the entire science of 

 hydromechanics, from the preliminary con- 

 cepts up to his vortex atom theory of mat- 

 ter, has been illuminated and extended by 

 his unrivalled fertility. 



"When we turn to the more intricate 

 branch of the subject which deals with the 

 motion of viscous fluids, or with the motion 

 of solids in such fluids, it appears that the 

 progress of the century is less marked, but 

 still very noteworthy. This branch is 

 closely related to the theory of elasticity, 

 and goes back naturally to the early re- 

 searches of Navier, Poisson and Saint- 

 Venant ; but the revival of interest in this, 

 as well as in the less intricate branch of the 

 subject, seems to date from the fi-uitful 

 memoir f of Stokes, of 1845, and from his 

 report to the British Association for the 

 Advancement of Science of 1846. Since 

 then many interesting and useful problems 

 relative to the flow of viscous fluids and to 



*'0n vortex motion,' 1867. Transactions of the 

 Eoyal Society of Edinburgh, Vol. XXV. 

 f Cited above. 



