52 ORIGIN AND SIGNIFICANCE 



and therefore ten equations, involving in space of three 

 dimensions fifteen variable co-ordinates. But of these 

 fifteen, six must remain arbitrary, if the system of five 

 points is to admit of free movement and rotation, and 

 thus the ten equations can determine only nine co-ordi- 

 nates as functions of the six variables. With six points 

 we obtain fifteen equations for twelve quantities, with 

 seven points twenty-one equations for fifteen, and so 

 on. Now from n independent equations we can 

 determine n contained quantities, and if we have 

 more than n equations, the superfluous ones must be 

 deducible from the first n. Hence it follows that the 

 equations which subsist between the co-ordinates of 

 each pair of points of a solid body must have a special 

 character, seeing that, when in space of three dimen- 

 sions they are satisfied for nine pairs of points as 

 formed out of any five points, the equation for the tenth 

 pair follows by logical consequence. Thus our assump- 

 tion for the definition of solidity, becomes quite suffi- 

 cient to determine the kind of equations holding be- 

 tween the co-ordinates of two points rigidly connected. 

 Thirdly, the calculation must further be based on 

 the fact of a peculiar circumstance in the movement of 

 solid bodies, a fact so familiar to us that but for this 

 inquiry it might never have been thought of as some- 

 thing that need not be. When in our space of three 

 dimensions two points of a solid body are kept fixed, 



