286 Motion of a System of Bodies. 



= (A'sin.4/+A;'cos.4').(A;'sin.4' — A^cos.^-)^, or [(^ — A)sin.4'Cos.4' 

 ■\-g'{cos.^^—sm.^l)']x[{k'g - h'g' - k'k)sin. -^^-^-ih'ki-k'g' -h'h) 

 COS. 4'] = (A'sin.4'+^'cos. 4^).(A;'sin.-^- h'cos.-^)"^ ; put w=tan.4^, 

 ax\dwehave{h'u-\-k').{k'u-h'y + {{g-h)u+g'{l-u^))x{{k'k-\- 

 h'g'—k'g)u-\-h'h—k'g'-h'k)=0, [v). Since [v) is a cubic equa- 

 tion, it has (by the theory of equations,) at least one real root; •'. « 

 is a real quantity; •'.%)-■ becomes known, and thence tan.^ is found 

 by the second of [u); .'. & is known : having found 4^ and 6, we can 

 easily obtain 9, for multiplying the first of (r) by dm, taking the inte- 



M 



gral and putting Sx'y'dm = 0, we have tan.29= - 1^5 M and N be- 

 ing known rational functions of sin. 4', cos. 4/, sin. &, cos. d, g, h, etc. 

 .'. (p becomes known; hence the position of the axes of cc', y', z' is 

 determined so as to satisfy (p) ; and it may be observed that the 

 axes thus found are called the principal axes of the solid. It may 

 be observed, that M=tan. 4= the tangent of the angle made by the 

 axis of X with the line of common section of the planes x, y, x', y'; 

 but it is evident that (p) will exist if we change y' into z', and z' into 

 y', that is, if we change the plane x',y' into x',z', and x',z' into x'^y'; 

 .'. {v) will give another value of u, which will be the tangent of the 

 angle made by the axis of x with the line of common section of the 

 planes x, y, and x', z' ; and in a similar way it may be shown that (v) will 

 give another value of u, which will be the tangent of the angle made 

 by the axis of x with the line of common section of the planes x, y, 

 y',z'', .". the three roots of {v) are real, and they appertain, gener- 

 ally, only to one system of axes : hence a solid has, in general, but 

 one system of principal axes passing through any given point. 

 Again, if Sxydm=g'=0, Sxzdm=h'=0, Syzdm—k'=0, the axes 

 of X, y, z will be principal axes : in this case, every term of (v) will 

 =0, but (u) become sin. ^ cos. ^ (^ sin.^ -^ + A cos.^-^ — A;) = 0, 

 (g- — A)sin.d sin.-j' COS. -4=0, also Sx^y'dm—abg-{-a'b'h-\-a'V'k=0, 

 (it>); and (t) becomes Sz'^dm= g s\n.^6 sm.'^ -^-i- h s\n.^& cos.'^-^-^ 

 k COS.'' 6 = gc' + he' ■^ + kc"^, {x); also S(x''' ■}-y'''+z''')dm~S{x'^ 

 -\-y'^)dm-{-Sz'-dm=S{x'' +y'' -{-z^)dm=g+hik={^nce c^'+c'^-}- 

 e"'' = l,)e''{hi-k)-}-c'^{g-\-k)-\-c'''^{g-{-h)+gc^-^hc'^-i-kc"^,oThy 

 (x), S{x'^+y'^)dm=e''{h+k)-\-c'^-{g-\-k)+c"Hg+h); put A+^ 

 =A, g-^k^Ji, g-\-h=C, and we have S{x'^ -\-y'-)dm=c^A-\-c''''B 

 +c"^C, (y). It may be remarked, that the first member of (y) is 

 the moment of inertia relative to the axis of z', and that A is the 



