282 Motion of a System of Bodies. 



if (p=0,then cos. 9=1, a=a', l = h', c=c', .'. a'^-^-b^ -JrC^^l, {b); 



P 



if (p = -x, then cos. (p = 0, .'. aa'+bb'-{-cc'—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 the other, and the 

 angle which these lines make with each other, is manifestly equal to 

 that which the given lines make ; hence, (a), (b), (c) may be found 

 as before. 



(2.) To change the rectangular coordinates of a point referred to 

 one system of axes, to the rectangular coordinates of the same point, 

 when referred to any other system of axes, having the same origin. 



Let oc, y, z, denote the coordinates of the point when referred to the 

 first system, and x', y', z' its coordinates relative to the second sys- 

 tem, also let L, denote the stright line drawn from the origin to the 

 point ; and let a, b, c denote the cosines of the angles which the 

 axis of X, makes with the axes of x', y', z', repectively, and a', b', &, 

 the corresponding cosines for the axis of y, also, a'', b", cf' the co- 

 sines for the axis of z. It is evident that a;=the projection of L on 

 the axis of a; = the sum of the projections of a;', y', z' on the same 

 axis: and the same remarks apply to y and z with respect to the pro- 

 jections of x\ y', z', on their axes ; hence x=ax'-\-by'-{-cz', y=a'x' 

 •\-b'y'-\-c'z', z^=a"x'-\-b"y'-{-c"z',{d); in the same way aj^=the 

 sum of the projections of x, y, z, on the axis of x', and y', the sum 

 of the projections on the axis oft/', and so of 2'; hence x'=aoe 

 -\-a'y-{-a''z, y'=^bx-\-b'y-\-b"z, z'=cx-\-c'y-\-c"z, (e). It is evident 



by (6) and (c), and because the two systems of axes are rectangular, 

 that we shall have a2-f^^+c^ = l,a'2_|_6/2_j_^/2_i^Q//2^_^//2_^g//2=. J ^ 



ab^a'b'-{-a"b"=0, ac+aV+a'V'=0, bc4-b'c' i-b"c"=0, a^-^a'^ 

 + a"2 = l, 62+6'2+6"2 = i, c2+c'2+c''2 = l, aa'+bb'-{-c&=^0, 

 aa"-{-bb"+c&' = 0, aV+5^6''+cV' = 0, (/) ; it is evident that (/) 

 are only equivalent to six independent equations, so that three of the 

 nine cosines which they involve are indeterminate. Again, (since 

 by if), &c+6V-f6'V'=0, l-c^=c'^-\-c"^, l-b^=b'^-\-b"\) 

 we have b'^-\-c^-=b''^c' -2bc{bc + b'e'-{-b"c")=b^{l-e-) -{-c^ 

 {l-b'-) -^2bc{b'c' -\-b"c")=^b^ {c'^ +c"^)-^c^ {b'^- -\-b"^)-2bc{b'c' -\- 

 h"c")^bH''^ -2bcb'c' -\-c'^b'^- -{-bH"^ —2bcb"c" -\-c-b"''={bc' -b'cY 

 ■^{bc" -b"cY=^l-{b'c"^b"c'Y; but b'- +c'' = \-a^, .\a^' = 

 {b'c"-b"c'Y or a=b'c" -b"c' ; ■a\sob = a"c' -a'c", c=a'b"-a"b', 

 a'=b"c-bc",b'^ac"-a''c, c' = a"b-ab", a"=bc'-b'c, b"=a'c 

 — ac', c"=ab' —a'b, [g)', it may be observed that the equation a^ = 



