PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



255 



dt S^i 



= W2, a* ', ; ji 5— = 1112 ^2 l ' • 'Tl sTT- = m X' 



'1** 1' dt 8.r„ 



'" di da: 



n n 



d lY ,, d 8V „ d 8V 



d l\ 



d sv 



rf^S;?, — /Wl^ i; ^^g^^ 



= //io ^ 



2' '" dtdz 



d 8V „ 



n 71 



(H.) 



for the second group, 



</^8ai ~^' dt 8a.2 ^ 



dtdc, "' 





= 



n 



dtdb — ^ 



n 



(I.) 



and finally, for the last equation, 



dt SH 





(K.) 



By combining the equations (C.) with their differentials (H.), and with the re- 

 lation (F.), we deduced, in the foregoing number, the known equations of motion (3.) ; 

 and we are now to show the consistence of the same intermediate integrals (C.) with 

 the group of differentials (I.), which have been deduced from the final integrals. 



The first equation of the group (I.) may be developed thus : 



82 V 





82 V 



+ ^'2 



8^V 



+ . . . + ^' 



S^V 



n 8 a, 8z 



i n J 



(14.) 



+ ^'i8ai8zi "1" -^28^18^2 



and the others may be similarly developed. In order, therefore, to show that they 

 are satisfied by the group (C), it is suflicient to prove that the following equations 

 are true, 



(L.) 



the integer i receiving any value from 1 to w inclusive ; which may be shown at once, 

 and the required verification thereby be obtained, if we merely take the variation of 

 the relation (F.) with respect to the initial coordinates, as in the former verification 



2 l2 



