INSTRUMENTS ILLUSTRATING KINEMATICS. 69 



well as carried about; but the motion may be described as the sliding 

 of one plane upon another. Thus in each case the matter to be 

 studied is the sliding of one surface on another which it exactly fits. 

 For two surfaces to fit one another exactly, in all positions, they must 

 be either both spheres of the same size, or both planes ; and the 

 latter case is really included under the former, for a plane may be 

 regarded as a sphere whose radius has increased without limit. 

 Thus, if a piece of ice be 'made to slide about on the frozen sur- 

 face of a perfectly smooth pond, it is really rotating about a fixed 

 point at the centre of the earth ; for the frozen surface may be 

 regarded as part of an enormous sphere, having that point for 

 centre. And yet the motion cannot be practically distinguished 

 from that of sliding on a plane. 



In this latter case it is found that, excepting in the case of a 

 pure translation, there is at every instant a certain point which is 

 at rest, and about which as a centre the body is turning. This 

 point is called the instantaneous centre of rotation; it travels 

 about as the motion goes on, but at any instant its position is 

 perfectly definite. From this fact follows a very important con- 

 sequence; namely, that -every possible motion of a plane sliding 

 on a plane may be produced by the rolling of a curve in one plane 

 upon a curve in the other. The point of contact of the two 

 curves at any instant is the instantaneous centre at that instant. 

 The problems to be considered in this subject are thus of two 

 kinds : Given the curves of rolling to find the path described by 

 any point of the moving plane ; and, Given the paths described by 

 two points of the moving plane (enough to determine the motion) 

 to find the curves of rolling and the paths of all other points. An 

 important case of the first problem is that in which one circle rolls 

 on another, either inside or outside; the curves described by 

 points in the moving plane are used for the teeth of wheels. To 

 the second problem belongs the valuable and now rapidly increas- 

 ing theory of link-work, which, starting from the wonderful 

 discovery of an exact parallel motion by M. Peaucellier, has 



