276 PROFESSOU HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



which auxiliary quantities, although in one view they are functions of the twelve ex- 

 treme coordinates, are yet to be treated as constant in calculating- the three definite 

 integrals, or limits of sums of numerous small elements, 



The form (H^.), for the characteristic function of a binary system, may be re- 

 garded as a central or radical relation, which includes the whole theory of the motion 

 of such a system ; so that all the details of this motion may be deduced from it by 

 the application of our general method. But because the theory of binary systems 

 has been brought to great perfection already, by the labours of former writers, it 

 may suffice to give briefly here a few instances of such deduction. 



14. The form (H^.), for the characteristic function of a binary system involves 

 explicitly, when ^ is changed to its value (79)j the twelve quantities x,, y^^ z^^ a^, b^^ c^, 

 r r^^^ h H^ H^^ (besides the masses m^ m^ which are always considered as given ;) its 

 variation may therefore be thus expressed : 



8V 8V 8V 8V 8V 8V 



SV SV SV 8V 8V 8V ^ ' \ ') 



In this expression, if we put for abridgement 



X = J ^ s^' ^'t^'L ^e > (80.) 



we shall have 



_ = X (07, - «,), §^^ = ?^ {yn - b,i), Yr,='^ (^" - ^/')' I 



= X(c,-2j;J 



g^^ = X {a„ - X,;), 8^^ = >^ {b, - y,;), -g^^ 

 and if we put 



?„ = + \/2 K + m,) (/(r„) + ^j - J, ....... .(81.) 



the sign of the radical being determined by the same rule as that of f, we shall have 



8 V m^m^g 8 V — fn^m^gQ S V m^m^h 



8r Wj + Wg' ^Vq m^ + m^ ' 8^ m^ + m^* 



(N2.) 



besides, by the equations of condition (F.), (K^.), we have 



w = ^' • . m) 



and 



SV SV y^rdr VTT . VTT VTT 



fw,=m;='fT' ^H, + ^H, = SH (P2.) 



