Motion of a System of Bodies. 283 
{b’c’ —b’c')? gives a=-(b/c” — bc’), but the sign — does not apply ; 
for supposing the coordinates 2, y, z to coincide with 2’,y’, 2’, we 
have a=1, 6’/=1, c/=1, 6”=0, c’=0; having determined the 
sign of a, the signs of 6 and c are also determined asin (g), for a, 
6, c, are to be taken so as to make aa’+-bb’+cc’ (identically) =0; 
and in a similar way have the signs of a’ 6’, &c. been determined 
as in (g). 
2, y,z can be found in terms of x’, y’, z, in another manner: for io 
ity, and clearness, imagine (with La Place, Mec. Cel. Vol. 1, p. 58.) 
that the origin of the codrdinates is placed at the centre of the earth, 
that the planes of «, y,and x’, y’ are the ecliptic and equator respect- 
ively, that the axes of z, 2’ are drawn to the north poles of the ecliptic 
and equator severally. 
» Let J=the angle made by the axis of # and the radius draw to 
the vernal equinox, 9 +}=the angle made by the axis of y and the 
same radius, these angles being reckoned according to the order of the 
signs ; put Qs ote foe the angles which the same radius makes with 
the axes of 2’, y' respectively, these angles being reckoned according 
to the direction of the earth’s rotation about its axis; let =the ob- 
liquity of the ecliptic =the angle made by the axes of z, 2’. It is mani- 
that the sum of the projections of «, y, z on any straight line, 
equals the sum of the projections of w’,y’,2/ on the same line; for 
each of these sums equals the projection of L on that line. Let the 
two systems of codrdinates be projected on the line of the equinox- 
es, then (since the projections of z, z’ are each=0,) we have x Cos. 
P i : P 
+ cos. (5+4) =<’ cos.o+Y’ cos. (+ °] or (since cos. (+4) 
as P 
== —sin. 1, cos. (5 fol -- sin. %,) cos) — ysin. L=2/ cos.9—y 
sin. , (h.) Again, let the two systems be projected on the line of the 
solstices, then (since the projection of z,=0,) we have w cos. (5 3 +) 
+ycos..)=2 sin. }+y cos. )=the sum of the projections of x, y, z; 
also 2’ sin. 9 cos. =the projection of «’, for it evidently equals the 
projection of 2’ on the line of common section of the solstitial colure 
(or plane of z, 2’,) and equator, projected on the line of the solstices; 
the first of these projections=2’ cos. g ~?} =2’ sin. g, and this 
