138 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



culty were not occasioned by the necessity of varying the lower limit q, which de- 

 pends on those two elements, and by the circumstance that at this lower limit the 

 coefficients 0^ Cl^ become infinite. However, the relation (202.) shows that we may 

 express this combination {r, u} as follows : 



{'">''} =J^fJ' ^rdr + ^fj' n^dr, (203.) 



r, being an auxiliary and arbitrary quantity, which cannot really affect the result, 

 but may be made to facilitate the calculation ; or in other words, we may assign to 

 the distance r any arbitrary value, not varying for infinitesimal variations of », (^, 

 which may assist in calculating the value of the expression (199.). We may there- 

 fore suppose that the increase of distance r — ^ is small, and corresponds to a small 

 positive interval of time t — r, during which the distance r and its differential coeffi- 

 cient r' are constantly increasing; and then after the first moment r, the quantity 



e. = ^ (204.) 



will be constantly finite, positive, and decreasing, during the same interval, so that 

 its integral must be greater than if it had constantly its final value ; that is, 



t^r=^JJ ©^dr>(r-q)(d, (205.) 



Hence, although 0^ tends to infinity, yet (r — q) 0^ tends to zero, when by dimi- 

 nishing the interval we make r tend to q ; and therefore the following difference 



J, ^rdr--^^jj^ e,dr = -y^J^ [^-j,)Q,dr, . . (206.) 



will also tend to 0, and so will also its partial differential coefficient of the first order, 

 taken with respect to /a. We find therefore the following formula for {r, u}, (re- 

 membering that this combination has been shown to be independent of r,) 



{r,«}=,^j|j-^7^ e,rfr + -^y5^7; e,rfrj; . . . (207.) 



the sign ^ _ ^ implying that the limit is to be taken to which the expression tends 

 when r tends to q. In this last formula, as in (199.), the integral / Q^ dr may be 



considered as a known function of r, q, x, ^, or simply of r, q, z, if |U/ be eliminated 

 by the first condition (186.) ; and since it vanishes independently of « when r = ^, it 

 may be thus denoted : 



J^ &rdr = (p{r,q,K) - <p{q,q,K), (208.) 



the form of the function p depending on the law of attraction or repulsion. This 

 integral therefore, when considered as depending on » and jM/, by depending on k 

 and y, need not be varied with respect to «, in calculating {r, u\ by (207.), because 



