284 Motion of a System of Bodies. 



projected on the line of the solstices gives x' sin. 9 cos. & as above ; by- 

 changing x' intoy, 9 into ^ +?j we have y' cos. (pcos. ^=to the pro- 

 jection of y'; the projection of z',=;r'cos. [j—&j=z'sm.&', hence 



(x' sin. 9+1/^ cos. (p) cos. ^+2;'sin.^=the sum of the projections of 

 x', y',z', =the sum of the projections o^ x,y, z; .'. x s\n.-\^+y cos. 

 4,==(a7'sin. (p-fJ/'cos.(p) cos. ^ + 2' sin. ^, (i.) 



Lastly, let the systems be projected on the line of common section 

 of the plane z, z' and equator : this is easily effected by (i,) viz. by 

 changing x'. y', z' into x, y, z severally, and x, y into x\ y'; ^ into 9, 

 and 9 into 4, ; (observing that the sign of the term involving z must 



be changed, for its projections^ cos. ( — + ^j=— z sin. ^; hence a;' 



sin. 9+2/' cos.9 = (a; sin. 4-4-2/ cos.-.)>) cos.^ — z sin. &, (Jc.) Multiply 

 (h) by COS. 4, (i) by sin. 45 then add the products and (since cos.^ 

 ^+sin. 24=1,) we have x=x' (cos.^ sin. 4 sin. 9 + COS.4 cos. 

 (p)_j-j/'(cbs. ^sin.4cos. 9 — cos.4sin.9)4-z' sin.^ sin. 4? fZj; change 

 the multipliers into, - sin. 4 and cos. 4; then (as before',) y=x' (cos. 

 ^ cos. 4 sin. 9 - sin. 4 cos. 9) -f-^' (cos. ^ cos. 4 cos. 9 + sin. 4 sin. 9) 

 -fsr'sin.^ cos. 4, (m); substitutea;sin.4+2/cos. 4 as given by (i), in 

 (yfc); then (since 1 — cos. 2^ = sin. ^6,) we have by reduction z=z' 

 cos.&—y' sin. a cos. 9 — a;' sin. ^sin.9, (n). {I), [m), (n) agree with 

 the equations which La Place has given at p. 58. Vol. 1 of the Mec. 

 Cel. and if I am not greatly mistaken they have been found by a 

 much more simple method than his. 



Now, since a, b, c, &c. 4? 9? ^s remain the same for every point of 

 space when referred to the axes of x, y, z, and^', y',z', .'.by taking 

 the point in the axis of x', we have y' = 0, z'=0 ; hence {d) become 

 a:=ax', y=a'x', z=a"x', and (l), {m), (n) become :r=:?;''(cos.^ sin. 

 4. sin. 9 + COS. 4/ C0S.9), y=x'{cos.6 cos. 4 sin.9 — sin.4 cos. 9), 

 2;=— re' sin. & sin. 95-'-by comparing these values of x,y,z, a=cos. 

 ^ sin.4 sin.94-cos.4 cos. 9, a'=cos. ^ cos. 4 sin. 9 — sin. 4 cos. 9, 

 a"= — sin.^ sin.9, in a similar way 6 = cos. ^ sin. 4 cos. 9 — cos. -4 

 sin. 9, 6'=cos. ^ COS. 4 cos. 9 4-sin. 4 sin.9, h"= —sin. ^cos. 9,c= 

 sin. d sin. 4, c'=sin. fl cos. 4, c''=cos. &, (0). 

 (3). Supposing the same construction and notation as before, \eix,y,z, 

 x\ y\ z'y denote the rectangular coordinates of any element cfm, of any 



