137-138] MODIFICATION OF LAGRANGE s EQUATIONS. 211 



for shortness, & = p/c^ + p/e a/ + . . . \ 



h/V+... } (19), 



J 



the formula (17) becomes 



(20). 



Selecting the terms in A^, A^ from the expression to be 

 integrated, we have 



i 7 2 y- 



! ^ % &quot;&quot;dft dq l 



Hence by a partial integration, remembering that A^, A^ 2 , ... 

 vanish at both limits, and equating to zero the coefficients of 

 Ag a , A&amp;lt;/o,... which remain under the integral sign, we obtain 

 the first of the following symmetrical system of equations : 



d d& d& dK 



T* ^-^ + (1,2)^., + (1,3)^+... + j- 



dt dq l dq l dq l 



d d& d& dK 



d df& d 1 . dK 



It d ~d + (3 1} + (3 2) + + 



We have here introduced the notation 



(r }S ) = d / s - d ^ (22), 



dq r dq s 



and it is important to notice that (r, s) = (s, r). 



138. The foregoing investigation has been adopted as leading 

 directly, and in conformity with our previous work, to the desired 

 result ; but it may be worth while to give another treatment of 

 the question, which will bring out more fully the connection with 

 the theory of gyrostatic systems, and the method of ignoration 

 of coordinates. 



* These equations were first given in a paper by Sir W. Thomson, &quot;On the 

 Motion of Rigid Solids in a Liquid circulating irrotationally through perforations 

 in them or in a Fixed Solid,&quot; Phil. Mag. , May 1873. See also C. Neumann, Hydro- 

 dynamische Untersuchungen (1883). 



142 



