Motion of a System of Bodies. 283 



{h'c" — h"c'Y givesa — ±(6V — 6^V), but the sign — does not apply; 

 for supposing the coordinates x, y, z to coincide with x\y\ z', we 

 have = 1, 6' = 1, c''=l, 6" = 0, c' = 0; having determined the 

 sign of a, the signs of b and c are also determined as in {g), for a, 

 b, c, are to be taken so as to make aa'-{-bb'-^cc' (identically) =0; 

 and in a similar way have the signs of a' b', Sec. been determined 

 as in [g). 



X, i/,z can be found in terms of a;', y', z, in another manner : for brev- 

 ity, and clearness, imagine (with La Place, Mec. Cel. Vol. 1, p. 58.) 

 that the origin of the coordinates is placed at the centre of the earth, 

 that the planes of x, y, and x',y' are the ecliptic and equator respect- 

 ively, that the axes of z, z' are drawn to the north poles of the ecliptic 

 and equator severally. 



Let •|'=the angle made by the axis of x and the radius drawn to 



P 



the vernal equinox, ~^-\-\' — \!i\Q angle made by the axis of y and the 



same radius, these angles being reckoned according to the order of the 



P 



signs; put 9, '2+9 for the angles which the same radius makes with 



the axes of ^r', ?/' 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, z' . It is mani- 

 fest that the sum of the projections of x, y, z on any straight line, 

 equals the sum of the projections of x', y', z' on the same hne ; for 

 each of these sums equals the projection of L on that line. Let the 

 two systems of coordinates be projected on the line of the equinox- 

 es, then (since the projections o^ z, z' are each = 0,) we have x cos. 



4-+?/ cos. (2 +^) =«'cos.(p+/cos. (2 + ^] °^ (^^"^^ ^°^* \2 "'■"^ ) 



/P \ . , 



= -sin.4'5 COS. I 2 +<?] = '- sin.cp,)'a;cos4' — 2/sin.4'=a;'cos.(p — y 



sin. (p, (A.) Again, let the two systems be projected on the line of the 

 solstices, then (since the projection of 2^,=0,) we have x cos. ( S'~4' ) 



-I-3/cos.4'=a?sin.4'-[-y COS. 4'=thesum of the projections of a:, y,z', 

 also x' sin. 9 cos. ^=the projection of x', for it evidently equals the 

 projection of x' on the line of common section of the solstitial colure 

 (or plane of z, z',) and equator, projected on the line of the solstices; 



the first of these projections=a?' cos. [^ — (f]=x' sin. 9, and this 



