INTRODUCTION. xxi 



and hence it follows, that the two rotations about OA and 

 OB can be replaced by the single one about OC. 



The correspondence between the solution of this prob- 

 lem and the principle embodied in the parallelogram 

 of force should be noticed. We see that rotations about 

 intersecting axes are compounded by the same rules as in- 

 tersecting forces. 



Composition of Rotations about Parallel Axes. We shall 

 now consider the case in which the two axes A and B, 

 about which the body receives small rotations a and j3, are 

 parallel. Divide the perpendicular distance ^between 

 the parallel axes A and B in the inverse proportion of 

 a and /3, and draw a line C parallel to A and B through 

 the point thus obtained. We shall show that a rotation 

 around C through an angle a + ]3 will be precisely 

 equivalent to the two given rotations. For consider any 

 point P in the plane at a perpendicular distance x from 

 C. Then the distances of P from A and from B are 

 respectively 



x + d-rt and x - d-^^ 

 a + 3 a + ]3' 



The effect of the rotations about A and B will, therefore, 

 be to raise P above the plane of A and B to an amount 



but rotation about C through an angle a + )3 would have 

 had precisely the same effect, and the same will be true 

 for every other point in the plane besides P. 



We thus see that rotations about parallel axes are com- 

 pounded by exactly the same laws as parallel forces. 



Translations* The rule for the composition of parallel 

 rotations would not apply if the two r otations were 



