36 GEOMETRY OF MOTION. [67. 



ratio of the angles of rotation. The sense of the segments 

 L^L, LL 2 , L^L^ must be taken into account as well as the 

 sense of the angles dO lt d9 2 , dO. The axis / lies between / x 

 and / 2 if dO ly d0 2 have the same sense ; otherwise it lies outside 

 the space between / x , / 2 on the side of the axis having the 

 greater angle. 



67. Two equal and opposite infinitely small rotations about 

 parallel axes produce an infinitely small translation equal to 

 Z : Z 2 dd (see Art. 64, Formula (3) ) directed at right angles 

 to the plane of the axes l lt / 2 . Conversely, an infinitely small 

 translation can always be replaced by two equal and opposite 

 infinitesimal rotations. 



68. An infinitesimal rotation of angle dO about an axis / 

 can be represented (Art. 57) by a rectilinear segment laid 

 off on / equal to d9, or, to avoid infinitesimal lengths, pro- 

 portional to dO. This geometrical symbol of an infinitesimal 

 rotation has all the characteristics of a vector (compare Arts. 45, 

 49) ; but it has one more which distinguishes it from the vector 

 representing a translation : it is localized, or attached to a 

 definite line ; for two equal and parallel rotations about different 

 axes do not represent the same thing. Such a localized vector 

 is called a rotor. 



69. The theory of rotors is of just as great importance in 

 mechanics as that of vectors (Art. 49). Angular velocities, 

 momenta, forces, all have for their geometrical representatives 

 rotors, i.e. rectilinear segments of definite direction, length, 

 sense, and situated on a definite line. 



The theory of the composition and resolution of rotors is a 

 matter of pure geometry ; it remains the same whatever the 

 rotor may represent. Thus we have seen in Art. 62, in the case 

 of infinitesimal rotations, that concurrent rotors are combined by 

 geometrical addition. The same rule holds for angular velocities, 

 momenta, and forces. In Art. 66 the rule for combining two 



