136 



KINETICS OF A RIGID BODY. 



[247. 



of the distances of the same point from the two planes. Thus, 

 if q be the distance of any point (x, y, z) from the axis of x y 

 we have <= L +& whence 



247. It follows, from the last article, that the moment of 

 inertia l x of a plane area, for any line perpendicular to its 

 plane, is 



/.=/,+/* 



if f y , I z are the moments of inertia of the area for any two 

 rectangular lines in the plane through the foot of the perpen- 

 dicular line. 



248. The problem of finding the moment of inertia of a given 

 mass for a line 1', when it is known for a parallel line 1, is of 



great importance. 



Let Hmq 2 ' be the moment of inertia of the 

 given mass for the line / (Fig. 30), ^mq l2> 

 that for a parallel line /' at the distance d 

 from /. The distances q, q 1 of any particle 

 m from /, /' form with d a triangle which 

 gives the relation 



S (q, d). 



Fig. 30. f* 



Multiplying by m, and summing over the whole mass M, we 

 find 



(q, d). 



Now the figure shows that the product qcos(q,d) in the 

 last term is the distance / of the particle m from a plane through 

 / at right angles to the plane determined by / and /'. We have, 

 therefore, 



(2) 



where the last term contains the moment of the first order 

 = Mp of the given mass M for the plane just me'ntioned. 



