700 Lord Rayleigh on Conformed Representation 



introduced, traversing the corner pieces through borings 

 making 45° with the previous ones. The model answered its 

 purpose to a certain extent, but the manipulation was not 

 convenient on account of the friction entailed as the wires 

 slip through the closely-fitting corner pieces. Possibly 

 with the aid of rollers an improved construction might be 

 arrived at. 



The material existence of the corner pieces in the model 

 suggests the consideration of a continuous two-dimensional 

 medium, say a lamina, whose deformation shall represent the 

 transformation. The lamina must be of such a character as 

 absolutely to preclude shearing. On the other hand, it must 

 admit of expansion and contraction equal in all (two-dimen- 

 sional) directions, and if the deformation is to persist 

 without the aid of applied forces, such expansion must be 

 unresisted. 



Since the deformation is now regarded as taking place 

 continuously, / in (1) must be supposed to be a function of 

 the time t as well as of £ + iw. We may write 



x + W = /(*> S + iv) ( 5 ) 



The component velocities it, v of the particle which at 

 time t occupies the position x, y are given by dxjdt, dy/dt, 

 so that 



u + iv = j t f(t, g+irj) (6) 



Between (5) and (6) ^ + iv may be eliminated ; u + iv then 

 becomes a function of t and of a? + zy, say 



u 



+iv = F(r, oc + iy) (7) 



The equation with which we started is of what is called in 

 Hydrodynamics the Lagrangian type. We follow the motion 

 of an individual particle. On the other hand, (7) is of the 

 Eulerian type, expressing the velocities to be found at any 

 time at a specified place. Keeping t fixed, i. e. taking, as it 

 were, an instantaneous view of the system, we see that it, v, 

 as given by (7), satisfy 



{d 2 /dx 2 +d 2 /dy 2 )(it, v) = 0, .... (8) 



equations which hold also for the irrotational motion of an 

 incompressible liquid. 



It is of interest to compare the present motion with that 

 of a highly viscous two-dimensional fluid, for which the 



