70 SCIENTIFIC APPARATUS. 



received an immense and most unexpected development at the 

 hands of Professor Sylvester, Mr. Hart, and Mr. A. B. Kempe. 



Passing now to the spherical form of this motion, we find that 

 the instantaneous centre of rotation (which is clearly equivalent to 

 an instantaneous axis perpendicular to the plane) is replaced by 

 an instantaneous axis passing through the common centre of the 

 moving spheres. In the same way the rolling of one curve on 

 another in the plane is replaced by the rolling of one cone upon 

 another, the two cones having a common vertex at the same centre. 



Analogous theorems have been proved for the most general 

 motion of a rigid body. It was shown by M. Chasles that this is 

 always similar to the motion of a corkscrew descending into a 

 cork ; that is to say, there is always a rotation about a certain 

 instantaneous axis, combined with translation along this axis. The 

 amount of translation per unit of rotation is called the pitch of the 

 screw. The instantaneous screw moves about as the motion goes 

 on, but at any given instant it is perfectly definite in position and 

 pitch. And any motion whatever of a rigid body may be pro- 

 duced by the rolling and sliding of one surface on another, both 

 surfaces being produced by the motion -of straight lines. This 

 crowning theorem in the geometry of motion is due to Professor 

 Cayley. The laws of combination of screw motions have been 

 investigated by Dr. Ball. 



Thus, proceeding gradually from the more simple to the more 

 complex, we have been able to describe every change in the 

 position of a body. It remains only to describe changes of size 

 and shape. Of these there are three kinds, but they are all 

 included under the same name strains. We may have, first, a 

 change of size without any change of shape, a uniform dilatation or 

 1 contraction of the whole body in all directions, such as happens 

 to a sphere of metal when it is heated or cooled. Next, we may 

 have an elongation or contraction in one direction only, all lines 

 of this body pointing in this direction being increased or diminished 

 in the same ratio ; such as would happen to a rod six feet long 



