GEOMETRY OF MOTION. 



[60. 



60. It is to be noticed that / x and / 2 are here regarded as 

 lines of the rigid body ; and while / x coincides with the position 

 of the first axis of rotation in space, the second axis of rotation in 

 space has the position l\, and not / 2 . It follows that, in general, 

 the order of the two rotations is not indifferent. But by repeat- 

 ing the construction, any number of rotations taken in a definite 

 order can be combined into a single rotation provided every axis 

 intersects the axis of the resultant of all preceding rotations. 



61. Again, in finding / from / x and / 2 , the positions of the 

 axes in the rigid body, as we did in Art. 59, the angle J^ is to 

 be applied to the plane /j/ 2 at /j in its proper sense, i.e. on that 

 side towards which the rotation about / x takes place ; but ^# 2 at 

 / 2 is to be applied to this plane in the opposite sense. If, 

 however, we wish to construct / from the absolute positions of 

 the axes of rotation in space, / : and /' 2 , we have to use 

 and +-|0 2 . 



62. In the case of two infinitely small rotations, say dQ^ and 

 d6y about intersecting axes / x , / 2 , the construction gains 

 remarkable simplicity. The resulting axis / falls into the plane 



of the given axes. 



Substituting d6 for sin# and 

 for cos#, the equations of Art. 59 

 assume the form 



(2'} 



sin (// 2 ) _ sin (/]/ 2 ) 

 i 



Fig. 18. 

 components d0 1 and 



These equations show that dO can be 

 found by geometrically adding the 

 rotors (Art. 57) representing the rota- 

 tions dO-L and dO%. In other words, the 

 s (or lengths proportional to them) being 



laid off on their respective axes (Fig. 18), the resultant rotation 



