BRACHISTOCHRONES 99 



The preparation of the great Treatise on Natural Philosophy led his 

 mind into various dynamical problems, such as central forces, the hodograph, 

 and the theory of Action. On these he communicated short notes to the 

 Messenger of Mathematics and to the Proc. R. S. E. His greatest effort 

 at this time was, however, his paper " On the Application of Hamilton's 

 Characteristic Function to special Cases of Constraint." In it he showed how 

 brachistochrones or paths of shortest time were to be discussed by the same 

 general method which Hamilton had applied to the theory of Action. Most 

 of the investigation was embodied in the second edition of Tait and Steele's 

 Dynamics of a Particle. Before publishing the paper Tait took the precaution 

 of asking Cayley if he had been forestalled. Cayley replied : 



" I have only attended to the direct problem of Dynamics, to find the motion 

 of a system under given circumstances, whereas the question of brachistochrones 

 belongs of course to the inverse one... and I really hardly know anything about 

 it. My impression is that the subject is new." 



In his address to Section A of the British Association in 1870 Clerk 

 Maxwell, when referring to the rival theories of light, said 



" To understand the true relation of these theories in that part of the field where 

 they seem equally applicable we must look at them in the light which Hamilton has 

 thrown upon them by his discovery that to every brachistochrone problem there 

 corresponds a problem of free motion, involving different velocities and times, but 

 resulting in the same geometrical path. Professor Tait has written a very interesting 

 paper on this subject 1 ." 



Now this discovery which Maxwell ascribes to Hamilton was really made 

 by Tait in the paper under discussion. Maxwell was usually very accurate in 

 his history, and we can imagine the glee with which Tait found his friend 

 tripping. He would by some merry joke make fun of Maxwell's momentary 

 deviation from the lines of historic truth. Accordingly on July 14, 1871, 

 Maxwell apologised in quaint fashion on an unsigned post card as follows : 



" O T' Total ignorance of H and imperfect remembrance of T' in Trans. R. S. E. 

 caused to suppose that H in his optical studies had made the statement in the 



form of a germ which T' hatched. I now perceive that T' sat on his own egg, but 

 as his cackle about it was very subdued compared with some other incubators, I was 

 not aware of its origin when I spoke to B. A. When I examined hastily H on 

 Rays I expected to find far more than was there. But the good of H is not in 

 what he has done but in the work (not nearly half done) which he makes other 

 people do. But to understand him you should look him up, and go through all 

 1 See Maxwell's Scientific Papers, Vol. n, p. 228. 



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