XXXI 



Thus the motnental ellipsoid is of great use in the discussion of 

 moments of inertia, in representing the motion of a body round a 

 fixed point when there are no impressed forces, and in other questions 

 in dynamics. Again, two non-homogeneous (or three homogeneous) 

 variables can be regarded as the co-ordinates of a point in a two- 

 dimensional geometry, such as that of a plane or the surface of a 

 sphere or any analytical surface; and any equation among the co- 

 ordinates is then interpreted as representing a curve (or curves, or 

 portion of a curve or curves) upon the surface. Similarly, when 

 there are three non-homogeneous (or four homogeneous) variables, they 

 can be regarded as the co-ordinates of a point in a three-dimensional 

 geometry, such as that of ordinary space ; corresponding to an equa- 

 tion among the variables, there is a surface (or surfaces) in space ; 

 corresponding to two independent equations among the variables, 

 there is a curve (or curves) in space; and corresponding to three 

 independent equations, there is a point (or points) in space. 

 In such cases the analytical relations can often, with great advan- 

 tage, be exhibited as geometrical properties. When the number of 

 non-homogeneous co-ordinates is greater than three (or the number 

 of homogeneous co-ordinates is greater than four), the circumstances 

 have greater need of such a representation, while there is a greater 

 difficulty in constructing some geometrical illustration ; and then it 

 can be obtained in a corresponding form only by the idea of a space 

 of the proper number of dimensions. To secure the possibility of 

 such a representation, it is necessary to evolve the geometry of 

 multiple space. 



For example, there are four single theta-f unctions, and their squares 

 are connected by linear homogeneous relations. In order to obtain 

 other properties of the functions themselves, it is convenient to regard 

 them as homogeneous co-ordinates of a point in (ordinary) space; the 

 amplitude in space that then is to be selected is the quadri-quadric 

 tortuous curve represented by those linear relations, viz., the curve 

 which is common to two quadric cylinders with intersecting axes. 

 Similarly there are sixteen double theta-functiona, with corresponding 

 linear relations among their squares. The associated geometry is 

 fifteen-dimensional ; the manifoldness in this space to be selected for 

 the discussion of the properties is the quadri-quadric two-dimensional 

 amplitude common to thirteen quadric hyper-cylinders. 



An initial difficulty in the construction of an analytical geometry 

 of n-dimensions is the expression of an amplitude of less than w 1 

 dimensions by means of equations that shall represent the complete 

 amplitude, and nothing besides the amplitude. It occurs in ordinary 

 solid geometry, the difficulty there being to obtain the expression of 

 a tortuous curve in space by means of equations that represent it 

 alone. For instance, a twisted cubic is frequently taken as the inter- 



