hii 
which may also be thus written, 
2 
a F 
map = mm'r—* (a’ — a), 
= mm'r— (a — a’), 
dé? 
and which give 
& da da eh, ae eee mm’ 
0=m (aa 72 + 556a) oe m (& de + ar) se 28 ——, 
6a, oa’ being any arbitrary infinitesimal variations of the vec- 
tors a, a’, and ér being the corresponding variation of r; 
because 
da(a’ — a) + (a’ — a) da + da‘(a — a’) + (a — a’) Oe’ 
= — (da — da) (a’ — a) — (a! — a) (8a — 8a) 
=—98.(@ —a?=0.r? = 2rdr = — 27°98 er. 
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: 
da da mm’ 
4Z.m (tao +375) -- 6s —— = 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, 2, y, z, making for 
this purpose 
a= te + jy + hz, a’ = ix’ + jy’ + kz’,.. (4) 
and eliminating the squares and products of the three imagi- 
nary units, 7,7, k, by the nine fundamental relations which were 
communicated to the Academy in 1843, namely, 
fe pe ee = ] 
ee 6 es L (5) 
jiz—hk hx —ijik=—j: J 
we are conducted, from the equation (3), to a well-known 
