8 The Rev. Brice Bronwin on the 



are both positive quantities. Also by hypothesis^ («) is ne- 

 gative, andy\ (b) positive : consequently in the equality 



the terms after h are, in the aggregate negative ; and in 



"Vi«r /i(*)2 "" 



the terms after — & are, in the aggregate positive. Hence, 

 by subtraction, 



— "JMv + v. >A -= A + & + a negative quantity 

 /i («) § /i ( 6 ) ; 1 J 



therefore, regarding only absolute numerical values, 



/,(«) + 77M* + 



But 6 — a is necessarily not less than 7z + Jc: consequently 



the condition which must be fulfilled whenever, as assumed 

 above, the doubtful roots are real : and this is the criterion of 

 Fourier. 

 Belfast, May 13, 1843. 



IV. On the Problem of Three Bodies. By the Rev. Brice 

 Bronwin*. 



LET M, m, and ml be three bodies acting upon each other 

 with forces as the reciprocal square of the distance ; let 

 x, y, and z be the coordinates of m, x 1 , y\ and z' those of m\ 

 both referred to M as their origin ; also let .2" = x' — x, 

 y" =y — y, 2" = j? — z be those of ml parallel to the former, 

 having m for their origin ; and let 



r= vV+j/ 2 + z 2 ,V''= W 2 +*/' 2 + 3 ,2 ,y = ^*" 2 +y' 2 + ^ 

 To abridge we shall make M + m = fx,, M + ml — /J, m + mi 



mfif 1 i H+m+m f =lS, lhemfQ = — — + — J ^ '- 



— jj, the differential equations of the motion of m about M are 

 r 



d^ dQ_ ^y dQ_ d*z dQ_ 

 dt* + dx ~ U ' dt* + 1$ "' dt* + 1z~- U ' 



• 



Communicated by the Author. 



