338 Sir Joseph Larmor 



measuring rod, which can make play in each element of extension 

 represented by SpSq and also be transferred from place to place: 

 and we can thereby impart or rather superpose metric quality on 

 the twofold which hitherto was purely positional or rather tactical. 

 The simplest plan is to follow Euclid, on the basis of the Pytha- 

 gorean theorem, and expressing absolute length according to 

 measuring rod by a symbol Ss, to impose a scale-relation of form 



8s^ = Sp^ + Sq^. 



But this metric cannot be applied consistently over a curved 

 surface, unless it is of the very special type that can be rolled out 

 flat: for other surfaces it is necessary to have the more general 

 type of relation 



S52 =fSp^ + ^g^pBq + hSq^ 



in which/, g, h are functions of the coordinates p, q. 



This specification of an imported metric thus determines the 

 surface: starting from a given small region of it, the form of the 

 surface in an outer threefold space can be gradually evolved by 

 prolongation so as to fit in with consistent application of this 

 metric. It is this idea of prolongation of a non-uniform manifold, 

 equivalent to its geometrical continuation within a flat one of 

 higher dimensions, that was Riemann's contribution to the ideas 

 of geometry. But the manifold itself is supposed to be given only 

 tactically or descriptively; and it is the metric that is imposed on 

 it that, by its demand for consistency in measurements, deter- 

 mines for it a form, as located in a higher flat manifold. This form 

 is expressed in detail analytically by the ' curvature ' at each place, 

 as specified by a set of functions (one in the case of a surface) of 



the successive gradients of the set /, g,h, If we keep the system 



self-contained by avoiding the immersion of it in a uniform 

 auxiliary manifold of higher dimensions, our resource is to deter- 

 mine the curvature as the simplest set of functions that are invariant 

 for local changes of coordinates. But, in order of evolution at any 

 rate, this invariance may be held to be only a derived idea. 



In any case the nature of the non-uniform manifold, as thus 

 determined by a metric imposed on formless space, has nothing 

 to do essentially with the coordinates p, q, ... to which it may 

 happen to be referred: it is settled by the algebraic form of the 

 functions/, g, h, ... expressed in terms of jj, q, ..., or in geometric 

 terms by the 'curvature' as so expressed. 



As a consequence, if we transform a surface from internal or 

 intrinsic coordinates p, q, to others p', q', which are assigned 

 functions of the former, so that we obtain 



§§2 = f'8p'^ + 2g'8p'8q' + h'8q'^ 

 and construct the surface implied in this new equation by the 



4 



