On an Equation in Laplace’s Mécanique Céleste. 39 
variable, we have not .mm’=X.m.3'm. In the first place, 
there is only one sign of integration in the first member, 
and there are two in the second; and each of these signs 
affects the whole of the quantities which follow it; conse- 
quently the operations which are made upon the same va- 
riables in the two members are not the same. Secondly, 
the number of bodies designated by m and m’, &c. are not 
the same in the two members. In the first member these: 
bodies are combined two by two without any of them 
being combined with itself; on the contrary, in the second 
member each body is combined with itself, as well as with 
all the others. In order, therefore, that the preceding 
equation be just, we must subtract from the second member 
all the terms which give the combination of each body with 
itself; this is expressly what Laplace has performed, p. 130, 
where he subtracts from 3.m. 3m een 
dx 
Thee 
Tt now only remains for me to prove that these two terms 
are those which were formed by the combination of each 
dz dy 
ah? dg 
If this variable be extended to different bodies, it would be 
necessary to designate them by different accents. Thus we 
the two terms 
++ &.mx. Sm. _. —S.my.X.m 
body with itself. a, y, belong to the same body. 
dx d ie 
find 2, y, =-5 -, belong exclusively to the body m, 2’, 
1, 2, &, to the body m’, and Cc ly 
WaT ppe eae Te to the body m, and soon. onsequentiy, 
we can neither give the accent to the variables attached to 
pad! ~ dy” 
; eb Se ‘ 
m, nox designate by 2”, y”, >-, —-, the variables attach 
ed to m’, m’”, &c. 
Now mm’ shows the combination of two different bodies, 
consequently .mm’ expresses the whole of the combina- 
tions of each body with all the others, we have then 
Semi! = mn! + mm" + m'm” + mm” 4+- &e. On the con- 
. . d dy 
trary, in the expression Xm.xXm. =; x and have not 
any accent; these two variables then belong to the same 
body. Besides, each has its particular sign of integration ; 
they ought then to be combined each to each, bearing the 
same accent, and not each of them with all the others. In 
the same manner we may ar on the yariables y, and 
wi 4 
