514 FOl RTH REPORT — 1834. 



volving, besides, n other constants called the masses, v/hich 

 measure, for a standard distance, the attractive or repulsive 

 energies. 



Denoting these w masses by ?Hi m^.,.m„^ and their 3 n rectan- 

 gular coordinates by x^ y^ s, a-„y„»„, and also the 3 n 



component accelerations, or second differential coefficients of 

 these coordinates, taken with respect to the time, by x" ^ y'\ 

 s^'\....x"„ij'„z"„^ he adopts Lagrange's statement of this pro- 

 blem ; namely, a formula of the following kind, 



2 . m {x" ^x + ifly + ^" 8^) = 8 U, . . (1.) 



in which U is the sum of the products of the masses, taken two 

 by two, and then mvdtiplied by each other and by certain func- 

 tions of their mutual distances, such that their first derived 

 functions express the laws of their mutual repulsion, being 

 negative in the case of attraction. Thus, for the solar system, 

 each product of two masses is to be multiplied by the reci- 

 procal of their distance, and the results are to be added in 

 order to compose the function U. 



Mr. Hamilton next multiplies this formula of Lagrange by 

 the element of the time d t, and integrates from the time o to 

 the time t, considering the time and its element as not subject 

 at present to the variation 8, He denotes the initial values, 

 or values at the time o, of the coordinates x y z, and of their 

 first diflPerential coeflicients x^ y' z', hy a b c and «' b' c' ; and 

 thus he obtains, from Lagrange's formula (1.), this other im- 

 portant formula, 



2 . m {x'^x - ana + y' ^ y - b'hb -i- ;i'8^ - c'Sc) = 8S, (2.) 



S being the definite integral 



s =X {^ + ^- 1 ^-"^ + ^' '^ ^"^} '^^- • ^^'^ 



. If the known equations of motion, of the forms 



mi x"i = . — , nii y"i — ^ — , mt &"i = y- • • (4.) 

 Xi yi Zi 



had been completely integrated, they wovdd give the 3 n coor- 

 dinates X y z, and therefore also S, as a function of the time t, 

 the masses /«,..,?«„, and the 6n initial constants a b c a' b' c'; 

 so that, by eliminating the 3 n initial components of velocities 

 a' b' c' we should in general obtain a relation between the 

 7 n + 2 quantities S, t, m, x, y, z, a, b, c, which would give S as 

 a function of the time, the masses, and the final and initial co- 

 ordinates. We do not yet know the form of this last function, 



