137-138] MODIFICATION OF LAGRANGE's EQUATIONS. 211 



for shortness, ' fa = p/c^ + p/e'a/ + . . . 



.(19), 



the formula (17) becomes 



...... (20). 



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

 integrated, we have 



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

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

 A^ 1? Af/ 2 ,... which remain under the integral sign, we obtain 

 the first of the following symmetrical system of equations : 



d d d . dK 



d 



We have here introduced the notation 



and it is important to notice that (r, 5) = (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, "On the 

 Motion of Eigid Solids in a Liquid circulating irrotationally through perforations 

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

 dynamische Untersuchungen (1883). 



142 



