PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS, 

 in which, by the identical equation (56.), 



S^x. 



we have therefore 





^♦J,. "^ Sw« S« §OT^ 8« Sot,. 8^ J' 



8H, 



Sot 



S«x. 

 t 



I BJ 6 •CT 



109 



(62.) 



(63.) 





and -r- ^ may be found from this, by merely changing / to ^ : so that 



(r,u)i,il \8>j^S^„8^^ S,,^8t!r^8W S>,^ + VS ,,^ S>,J^^ ^'J, ^\ S^/"S^ 



■8 X. 8x 



Sx 

 8 X 8 X. 



Sx S X. 





\ (64.) 



and similarly, 



(r)i\^'sr^dt 8>j^ 8c7^ f/^ 8>jy 



8 X. 8^ X 



^ 3n,Sn|/8x^ 8^x. _ 8x. S^^ XgH,, / ^ '^j 



8x 8^x. 



Sta- 8« 8>j 



8x 8XA g2fj^ 



■ST 8)j Sot d'UT 8>3 / 8>j 



«< 'r r u 'r 'u 



8x 8x. 8x. 8x 



) 



■ (65.) 



/OX. 0X^._ OX^ OXA g-2H^ /l^_^__^i .^"jiULl 



\8 -cr^ 8 M 8 CT 8 jj / Sot 8m \8 ■bt 8ct 8 •or Sot / 8 « 8 « f 



r 'a r 'm u 'r r u r u 'u 'r •' 



Adding, therefore, the two last expressions, and making the reductions which pre- 

 sent themselves, we find, by (60.), 



dt h-f 



3n / fi 



2 (A 



(«) 1 v /, 



-f B ^ 



in which 

 A 



i,s 



^^3n,8x 

 r)l V 



8^X. 



8,, 



8 X. 8*x 



)' 



(D>.) 



8x. 8«x 8x S^x. X 



dtff 8cr 8m Sw 8 ■cr 8m/' 



(r)i\8M8i3"8'57 SmSctSw ' Gtff ocrOM 



N ' 'r u r 'r u r r u 'r 



,.W ^ 3n . 8x^ 8^x, 8x. 8^x 8x. S^^x 8x S^x. \ 



»,* (r)! \8cr^ 8rj^8>j^ 8ot^ 8)j^8rj^ Sj?^ 8»j^8w^ S)j^ 8>j^ 8^/ ' 



(66.) 



and since this general form (D^) for t;«,,j contains no term independent of the dis- 



8 H 8 H 

 turbing quantities -y- ^, -g^, it is easy to infer from it the important consequence 



already mentioned, namely, that the coefficients a^^^, in the differentials (B'.) of the 

 elements, may be expressed as functions of those elements alone, not explicitly in- 

 volving the time. 



