203 
Mortron oF N Boptgs. 
By ArtTHuR 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) ¢=¢7 
where @ is a matrix which transforms n determining points Of a reference space 
. of order n—1 into the positions of the n bodies, and 7 is a self-conjugate 
matrix, depending solely upon the ratios of the mutual reactions to the corres- 
ponding mutual distances. 
The matrix ¢ is of order »—1, if the motion of the bodies is within the 
reference space, and 9’, the conjugate of ¢, 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 »—I/’st order the matrix ¢ must be of 
the same order, but must annul all directions outside of the reference space. 
The reduced equations of motion are, 
(2) (P+ W) v—! (b—W) =2 (W— Yr — ry), 
(3) W=rp— pr, 
where y= #¢’@, 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) De (h)— yr — rh) = mh + pe + Wr —eW. 
Rose Polytechnic Institute, 
Terre Haute, Ind. 
