368 Prof. Clausius on the Connexion of the Second Proposition 



initial point*. If we then retain the letters i and E for the du- 

 ration of the motion and the energy respectively, and denote by 

 V the velocity of the point, and if we further, with variable quan- 

 tities, indicate their mean value by putting a horizontal stroke 

 over them, we can give to Hamilton''s equation as quoted under 

 (2) the following form : — 



h{m7H)=ih'E (2fl) 



In first forming my equation t I wrote it thus : — 



SU= I aJ^-f /???'51og 2 (3) 



But I did not there employ the letter U as a universally valid 

 svmbol for the ergal, but said explicitly that it only denoted the 

 ergal/or the original motion. In a subsequent memoir J, in order 

 to show the difference very clearly, I wrote the equation in the 

 following form : — 



-r-hx-\- -^— Sv+ -j-hz= — Si/*^ -{- mv^S log t : . (3^) 

 dw dy ^ dz 2 o ^ \ j 



and I added, ^^ The sura 



dJJ ^ dV 5, , dV 5, 



dx dy "^ dz 



cannot at once be regarded as the variation of the ergal, and 

 hence, if the signification of U be extended so that it shall re- 

 present the ergal not only for the original, but also for the 

 altered motion, cannot at once be denoted by SU/^ 



If we assume for example the above-mentioned case, that the 

 function U contains, besides the coordinates, also a quantity c 

 which is constant with each motion, but may change its value 

 from one motion to another, then the above sum does not con- 

 tain that alteration of the ergal which is occasioned by the alte- 



* If the latter condition be not fulfilled, the difference 



\dt dt ^ dt /I \dt dt ^ dt /o 



(the indices and 1 signifying the initial and final values of the quantity) 

 must occur in the following equations. For our considerations, however, 

 the simpler form of the equations suffices, ^hich corresponds to the as- 

 sumption that the difference is =0. 



t Sitzungsber. der XiederrJiein. Ges.fiir Xafur- mid Heilkunde, 1870, 

 p. 174 ; and Phil. Mag. September 1871, P- 167. 



X " On the AppHcation of a 3Iechanical Equation advanced by me to the 

 Motion of a Material Point round a fixed Centre of Attraction, and of two 

 Material Points about each other," Phil. Mag. November 1871, p. 321 ; 

 XachricJiten der Gottinger Gesellschaft der JVissenschaften, 1871} p. 248; 

 Math. Ann. von Clebsch u. >'eumann, vol.iv. p. 232. 



