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Motion of N Bodies. 



By Arthur S. Hathaway. 



The relative motion of n bodies, in any order of space, and subject to any 

 law of mutual action, is given by 



(1) = 07T 



where <p is a matrix which transforms n determining points of a reference space 

 of order n — 1 into the positions of the n bodies, and n is a self-conjugate 

 matrix, depending solely upon the ratios of the mutual reactions to the corres- 

 ponding mutual distances. 



The matrix <j> is of order n — 1, if the motion of the bodies is within the 

 reference space, and (f> / , the conjugate of <j>, annuls every direction of the refer- 

 ence space exterior to the space of the moving bodies. If the space which con- 

 tains the moving bodies be greater than n — l'st order the matrix (j> must be of 

 the same order, but must annul all directions outside of the reference space. 



The reduced equations of motion are, 



(2) (i> 4- W) V ~ X (t — W) = 2 (i> — i'K — 7T1JJ), 



(3) W = wi>—pr, 



where i/' =r / 0, a function of the mutual distances, and W is a skew conjugate 

 matrix, whose elements are to be found from the quadratic equations be- 

 tween them in (2), and thence substituted in the remaining equations of (2) and 

 in (3), giving a certain number of reduced equations of second and third order. 

 Another equation which is linear in the elements of W enables us to find 

 the reduced equations in third and fourth orders, 



(4) D t ( 'i> — inr — 7np) = nip f i}tt -\- Wtt — ttW. 



Rose Polytechnic Institute, 

 Terre Haute, Ind. 



