284 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



in which we are to observe that 



^ = y|2 + ^^ + C' + n/«' + (^'' + 7\ 1 



= ± x/(i - ^? + (^ - ^Y + (C - y)'^./ 



(M3.) 



By this comparison we are brought back to the general integral equations of the 

 relative motion of a binary system, (89.) and (90.) ; but we have now the following 

 particular values for the coefficients A, B, C : 



\ Iw , \ lia -r> l8ty ^ 18wj, iSw ,^,„ , 



rSo-'xOT' tot' VqIg- * t ot ^ ^ ' 



and with respect to the three partial differential coefficients, g— , g— , g— , we have the 

 following relation between them : 



^8T + '^87+^8T = ¥' (O'-) 



the function w being homogeneous of the dimension \ with respect to the three quan- 

 tities a, ff, r ; we have also, by (P.), 



So- V a * ^/ — COS o/ 8 T V a cos u^ — ^/ ^ '' 



and therefore 



g7g7 — 0-2 -t2' Vsvj + VsT; +t — (t^-t^' v^-^ 



from which may be deduced the following remarkable expressions : 



/8'rw 8^x2 4jiA ju, 1 



Is^ + 87 ) ~~ 7+1^ ~ "aT^ 



> (R3.) 



(8tt; 8w)\2 4 ft jtx. 



8t8o-/ <r — Ta*„ 



These expressions will be found to be important in the application of the present me- 

 thod to the theory of elliptic motion. 



16. We shall not enter, on this occasion, into any details of such application ; but 

 we may remark, that the circumstance of the characteristic function involving only 

 the elliptic chord and the sum of the extreme radii, (besides the mean distance and 

 the sum of the masses,) affords, by our general method, a new proof of the well- 

 known theorem that the elliptic time also depends on the same chord and sum of 

 radii ; and gives a new expression for the law of this dependence, namely, 

 2 a^ 8 to 



' = T^- (S'-) 



We may remark also, that the same form of the characteristic function of elliptic 

 motion, conducts, by our general method, to the following curious, but not novel 

 property, of the ellipse, that if any two tangents be drawn to such a curve, from 

 any common point outside, these tangents subtend equal angles at one focus ; 



