284 Motion of a System of Bodies. 
projected on the line of the solstices gives x’ sin. p cos. as above; by 
: P 
changing 2’ into y’, ¢ into ato, we have y’ cos. cos. é=to the pro- 
jection of y’; the projection of 2’,=2/ cos. (F—.) =2’sin. 4; hence 
(2’ sin. 9+-4/ cos. ) cos. 6+-2’sin. é=the sum of the projections of 
2’, y’, 2’,=the sum of the projections of x,y, 23 .'. a sin.b+y cos. 
l= (2' sin. o+y’ cos.) cos. +2’ sin. 4, (2.) 
Lastly, let the systems be projected on the line of common section 
of the plane z, 2’ and equator : this is easily effected by (7,) viz. by 
changing 2’, y’, z’ into x, y, 2 severally, and 2, yinto 2’, y’; J into 9, 
and 9 into .) ;.(observing that the sign of the term involving z must 
i 
be changed, for its projection=z cos. (5 +4) =—2 sin. 6; hence 2’ 
sin. 9+ 9’ cos. o=(2 sin. L-+y cos.) cos. é—z sin. 6, (k.) Multiply 
(h) by cos. J, (i) by sin. J, then add the products and (since cos.” 
+sin. ?}=1,) we have v=z’ (cos.é sin. | sin. ¢+cos. cos. 
o)+y' (cos. é sin. | cos. p—cos. sin. g) +2’ sin.é sin. 1, (1) ; change 
the multipliers into, — sin.) and cos. }; then (as before,) y=.’ (cos. 
é cos. |. sin. p — sin. L cos. 9)+y" (cos. 4 cos. J cos. 9+sin. J sin. ¢) 
+2/sin.4 cos. |, (m); substitute x sin. )+-y cos. J as given by (2), in 
({k); then (since 1—cos. *6=sin. ?6,) we have by reduction z=2’ 
cos. d—y! sin. 6 cos. p—2’ sin. dsin.g, (n). (1), (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 neve been found oy a 
much more simple method than his. 
Now, since a, b, c, &c. 1, 0,4, remain the same for every point of 
space when referred to the axes of L,Y, Z, and 2x’, y’, 2’, .”. by taking 
the point in the axis of 2’, we have y’=0, 2’=0; hence (d) become 
xan’, y=a' x’, z=a"x', and (1), (m), (n) become 2 =2"(cos. 6 sin. 
} sin. 9+cos.] cos.¢), y=x’ (cos. 4 cos. sin. g—sin.) cos. 9), 
z= -—v2’ sin. é sin. 93.".-by comparing these values of x, Age acs 
4 sin. J sin. pt cos. + Cos. ?, a’=cos, é cos. J sin, o—sin. | cos. 9, 
a/’=—sin.é sin.?, in a similar way b=cos. 4 sin. + cos. o—cos.f . 
sin. 9, b'=cos. 4 cos. | cos. 9 +sin. L sin. 0, b= —sin. 6 cos. 9, C= 
sin. 4 sin. J, acisooe 6 cos. + seo 4, (0). 
(3).8 as before, let x, y, 2, 
Sey 2's ‘denote the roctengliad coordinates of any element dm, of any 
