PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



97 



(3.) 



d §T §T _ SU 



in which 



T=:i2.m(^2+y24.2'2) . (4,) 



For, from the equations (2.) or (1.), 







in which 





82' 



(5.) 



(6.) 



and 



^•"^K^ dt8ri,+y rf^8>,, + ^ rf/8^J 

 ■ / ,8^' , ,8y , ,8s!'\ 8T 



(7.) 



T being here considered as a function of the 6 n quantities of the forms ri' and 73, ob- 

 tained by introducing into its definition (4.), the values 



of = t] 



8 a; 



1 8 



♦Ji 



+ ^' 



8 a; 



2S 



»J2 



+ ... + '?' 



8 ^ 



3n Si 



, &C. 



'3n 



(8.) 



A different proof of this important transformation (3.) is given in the M^canique 

 Analytique. 



3. The function T being homogeneous of the second dimension with respect to the 

 quantities f]', must satisfy the condition 



8T 



2T = 2 .??' 



8V 



(9.) 



and since the variation of the same function T may evidently be expressed as follows, 



^8T . . . ST 



^ 



T=2(^^,' + ^S,), (10.) 



we see that this variation may be expressed in this other way, 



ST ST 



8T=2(,'S^-^S,) ■ • • ■ ("•) 



If then we put, for abridgement. 



8T 



8V, 



8T 



HT 



1' • • • 8V, 



= OT. 



3w 



3n' 



(12.) 



MDCCCXXXV. 



