9i56 CELESTIAL MECHANICS. J. vii. Q7 . 



the planes of a:" and z" and of y and z", the solution must 

 be equally applicable to the three intersections formed by 

 the plane of x and y with the three principal planes of the 

 body respectively : hence it follows that all tlie three^roots 

 of the equation are possible, and that they determine in all 

 cases the same system of three axes in the body: [although, 

 as Euler observes, it would be difficult to demonstrate, from 

 a direct consideration of the equation, that all its roots must 

 necessarily be possible.] 



348. Corollary. If O be the rotatory 

 inertia with respect to the axis z\ we shall 

 find C- A sin ^e sin V + E sin ^0 cos V + C 

 cos 'a. 



This may be shown by substituting the values of jc' and y' 

 in the expression C'=S(y«+y'2)D/7z[; or rather in C'=S 



j {x"^ -\-y"^ '\-z"^)—z'^ I Dm, which is equal to it, z'« being 



z=2f'^ cos ^9-{-y"^ sin ^9 cos ^(p+x''^ sin 25 sin ^(p; since 

 all the products of the cross multiplications vanish in the 



integral, and CzuSjx"^ (1— sin 25 sin 2^)+2/"2 (l-~sin«fi 



cos ^(p) + z"^ (1 — cos ^9) >Bm, whence, by adding to- 

 gether two of the coefficients, and subtracting the third, 

 as in article 345, we have sin 25 (sin ^(p — cos ^(p)+s'm ^9=. 

 sin ^9 (2 sin ^(p—l+l) and half this, or sin 2<p sin ^9, is the 

 coefficient of A ; hence we have, secondly, sin ^9 — sin ^(p 

 sin H=.sm^ cos ^<p; and thirdly, I — sin H cos ^(p — sin ^ 

 sin 2^=1 — sin 25— cos ^9 for the coefficient of C]. 



Scholium. The quantities sin H sin ^(p, sin ~9 cos ^<p, 

 and cos ^9, are the squares of the cosines of the angles made 

 by z' with x'\ y\ and z" : [for with respect to zf\ it is ob- 



