282 Motion of a System of Bodies. 
if o=0, then cos. p=1, a=a’, b=’, cmc’, °. a? +03+¢7=1, (b)5 
P. ; 
if 9= then cos. -=0, -”. aa’ + bb’ +-cce’=0, (c). 
If the given lines do not lie in the same plane,’ then through any 
point in one of them, draw a straight line parallel to thé other, and the 
angle which these’ lines make with each other, is manifestly equal to 
that which the given lines make; hence, (a), (6), (c) may be found 
as before.) >. ’ 
(2.) To change the rectangular codrdinates of a point referred to 
one system of axes, tothe rectangular codrdinates of the same point, 
when referred to any other system of axes,.having the same origin. 
Let 2, y, z, denote the codrdinates of the point when referred to the 
first system, and 2’, y’, z’ its codrdinates relative to the second sys- 
tem, also let L, denote the stright line drawn from the origin to the 
point; and let a,6,¢ denote the cosines of the angles which the 
axis of 2, makes with the axes of «’, y’, 2’, repectively, and a’, b’, c’, 
the corresponding cosines for the axis of y, also, a’, 6”, ¢” the co- 
sines for the axis of z. It is evident that =the projection of L on 
the axis of z=the sum of the projections of 2’, y’, z’ on the same 
axis: and the same remarks apply to y and z with respect to the pro- 
jections of 2’, y’, z’,on their axes; hence r=az’+by/+c2', y=a'a! 
+0'y/ +2’, za" a! -+-b’y'+-c’2', (d); in the same way a’=the 
sum of the projections of 2, y, z, on the axis of «’, and y/, the sum 
of the projections on the axis of y’, and so of 2’; hence a’=aar 
+a’y-a"z, y =ba+ b/y+- bz, 2 =co+c/y+e%z, (ce). Itisevident 
by (6) and (c), and because the two systems of axes are rectangular, 
that we shall have a?-b?-+c? =1, a/246?+¢=1, a/?-+b/2.1.¢2=1, 
ab-+-a'b!+-a!/b=0, ac--a'e'-La!’e’ =0, be +b'c! + be =0, a? +a’ 
ball, be +bA4b4=1, c&4e%+4+c/27=1, aa’+bb’+cc'=0, 
aa” +-bb"' +-cc'=0, a/al’ +b/b! + e'c! =0, (f); it is evident that (f) 
are only equivalent to six independent equations, so that three of the — 
nine cosines which they involve are indeterminate. Again, (since 
by (f), be+bc-+b/c'=0, 1—ct me? +e", 1—-b2 U2 +0",) 
we have b* +c? =b* +-¢% —2be(be + b/c! + bc”) =b3(1 —e*) Le? 
(1-53) — 2be( b/c’ -+.bc") = b?(e'2 4-02) 4.62 (b'2 5/2) — Qbe( b/c + 
bic! \==b? cf? — Qheb’e’ +07 b!? Lb%¢/2 — Qheb//c-+-2b//2=(be! —D/e)? 
+ (be" — bc)? =1— (b/c —be')?; but b2+-e2=1~—a2, +a? = 
(b/c! —b/'c’)? or a=b'c” — bc! also b=ae!—a’el’, cm alb’ —alV’, 
a =e ~ be", & =ac! —a''e, e=a''b — ab", a!’=be — be, Wat 
ac’, e = ab’ —a’b, (¢); it may be observed that the equation a? = 
