lii 



which may also be thus written, 



m -TTT = mm'r (a — a), 

 dr 



m' -r-g- = mm'r-^ (a — a'), 



and which give 



Sa, da' being any arbitrary infinitesimal variations of the vec- 

 tors a, a', and 8r being the corresponding variation of r ; 

 because 



da(a' - a) + {a' - a) da -^ da'{a - a') + (a - a')Sa' 



- _ (ga' - da) (a' - a) - (a' - a) (Sa' - Sa) 



= _ g . (a' _ a)' = 8 .r^ = 2r8r z= - 2r^S . r-^ 



And by extending this reasoning to any system of bodies, 



we deduce from the equation (1) this other formula, by which 



it may be replaced : 



/- d^a d^a ^ \ . ^^ mm' .„. 



is . m [daj-^ 4- ^Saj + 8S -^ = 0. (3) 



Although it is believed that this result (3), if regarded 

 merely as a symbolic form, is new, as well as the method by 

 which it has been here obtained ; yet if we transform it by the 

 introduction of rectangular coordinates, x, y, z, making for 

 this purpose 



a = ea; + jy + kz, a' = ix' + JV' + ^2', . . (4) 



and eliminating the squares and products of the three imagi- 

 nary units, i,j, k, by the nine fundamental relations which were 

 communicated to the Academy in 1843, namely, 

 ^•2 =/ = ^2 = - 1 ; 1 



ij z= kjk = i, ki -j; Y (5) 



ji — — k, kj — — i, ik — —j \ } 

 we are conducted, from the equation (3), to a well-known 



