254 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



diate integral system (C), (E.), or even from a part of that system, namely, from the 

 group (C), when combined with the equation (F.). For we thus obtain 



^1^1 — dthx^ — ^ 



8!v , Jiv_ 



1 dx\ ^^^hx.^x^ 



-f . . . 4- ^' 



n 8 j7j S<r„ 





+^'l 





r-Y 



+ "'+y'n 



g2V 



gay . . S^V . . . S^V 



S^TiS^i ^■*2 8a:i8«2 



+ . . . + ^' 



n Sl-j 82„ 



1 SV 8^V 



I •»»» 



1 8V 8^V 



OTj Sj^i 8a;^i *^ m^^x^txida^ 



] 8V 8^V 

 + ••• + " 



+ 

 + 



1 5V S^v 



m^ Sj^i 8^i8j/i 



1 SV 8*V 



+ 



+ 



1 SV 8^V 

 1 8V S^v 



+ 



m„ 8^„ 8x1 8^„ 



* * I in 



^n^J/n^^Jl/n 



, . . . 1 8V 8^V 



(11.) 



that is, we obtain 



"^lA^I^: • (12.) 



And in like manner we might deduce, by differentiation, from the integrals (C.) and 

 from (F.) all the other known differential equations of motion, of the second order, 

 contained in the set marked (3.) ; or, more concisely, we may deduce at once the 

 formula (1.), which contains all those known equations, by observing that the inter- 

 mediate integrals (C), when combined with the relation (F.), give 



8V 8 



Iz) 



L (13.) 



1 /8V 8 8V 8 8V 8 \ /8V. , SV ,SV.\ 



=2{ix^^+^ij~+hz~^2.-^^ (i^)V (-87) + C^)7 



5. Again, we were to show that our intermediate integral system, composed of the 

 equations (C.) and (E.), with the 3 n arbitrary constants a^, h^, q, . . . «„. fe„, c„, (and 

 involving also the auxiliary constant H,) is consistent with our final integral system 

 of equations (D.) and (E.), which contain 3 n other arbitrary constants, namely, 

 «'i, ^'i> ^15 . . . «'„, ^'„, c'„. The immediate differentials of the equations (C), (D.), (E.), 

 taken with respect to the time, are, for the first group. 



