28S Sir 'W. R. Hamilton o)i certain discontinuoiis Integrals. 



equations of hydrodj'namics the generality requisite to meet 

 every proposed instance of motion. In a Report on the Ana- 

 lytical Theory of Hydrodynamics, read at the third meeting of 

 the British Association, I mentioned that it was a desideratum 

 in this department of apphed mathematics, to determine under 

 what circumstances of the motion udx + vdiz + tvclz is an ex- 

 act differential. The conclusions 1 have now arrived at seem 

 to clear up this difficulty. 



Cambridge Observatory, Feb. 11, 1842. 



XLIII. 0?i certain discontinuous Integrals, connected with the 

 Development of the Radical which represents the Reciprocal 

 of the Distance between two Points. By William Rowan 

 Hamilton, LL.D., P.R.I. A., Member of several Scientific 

 Societies at Home and Abroad, Andrew's Professor of Astro- 

 nomy i?i the University of Didilin, and Royal Astronomer of 

 Ireland*. 



1. TT is well known that the radical 



-*- {\—%xp + a;2)-4, (1.) 



in which x and 1 may represent the radii vectores of two 

 points, while p represents the cosine of the angle between 

 those radii, and the radical represents therefore the recipro- 

 cal of the distance of the one point from the other, may be 

 developed in a series of the form 



Po + Pja- + Po.r^ + ... + P„A'" + ...; . . . (2.) 



the coefficients P„ being functions of p, and possessing many 

 known properties, among which we shall here employ the fol- 

 lowing only, 



'■.= M-(4;)'(^)"---- '-) 



the known notation of factorials being here used, according 

 to which 



1111 



t«J-" = T-T--3 -IT (^•) 



It is proposed to express the sum of the first n terms of the 

 development (2.), which may be thus denoted, 



T^-\ \\ .r" = P„ + P, X + P, x^- + ... + P„_. ^''-'. (5.) 

 2. In general, by Taylor's theorem, 



fip+9) = \)Z [0]-'' r (^Yfip)-^ ■ ' ' (6.) 



• Communicated by the Author. 



