51G FOVRTII RKPORT — 1834. 



which may vigorously l)e thus transformed, by the help of the 

 equations (o.), 



S beino- any arbitrary function of the same quantities, f, m, x, 

 7/, 1^, a°b, c, supposed only to vanish (like S) at the origin of 

 time. If this arbitrary function S, be so chosen as to be an 

 approximate value of the sought function S, (and it is always 

 easy so to choose it,) then the two definite integrals in the for- 

 mula (8.) are small, but the second is in general much smaller 

 than the first; it may, therefore, be neglected m passing to a 

 second approximation, and in calculating the first definite in- 

 tegral, the following approximate forms of the equations (b.) 

 may be used, 



^i=_„,a',^4^=-.«i'//^=--c' . (9.) 



^a 2 6 ^^ 



In this manner, a first approximation may be successively and 

 indefinitely corrected. And for the practical perfection of the 

 method, nothing further seems to be required, except to make 

 this process of correction more easy and rapid in its appli- 

 cations. _, ,1 . ^^,„ 

 Professor Hamilton has written two Essays on this new 

 method in Dynamics, and one of them is already pnnted in 

 the second part of the Philosophical Transactions {o^ hondoTx) 

 for 1831. The method did not at first present itseli to him 

 under quite so simple a form. He used at first a Character- 

 istic Function Y, more closely analogous to that optical func- 

 tion which he had discovered, and had denoted by the same 

 letter, in his Theori/ of Systems of Rays. In both optics and 

 dynamics, this function was the quantity called^c^^OM, consi- 

 dered as depending (chiefly) on the final and mitial coordinates. 

 But when this Action-Function was employed in dynamics, it 

 involved an auxihary quantity H, namely the known constant 

 part in the expression of half the living force of a system; and 

 many troublesome eliminations were required m consequence, 

 which are avoided by the new form of the method. 



Mr Hamilton thinks it worth while, however, to point out 

 briefly a new property of this constant H, which suggests a 

 new manner of expressing the differential and integral equa- 

 tion« of motion of an attracting or repelling system. It is otten 



