698 Lord Rayleigh on Conformed Representation 



upper limit, the 4-lobed form requires least pressure for its 

 maintenance. Thus the true curve conuecting pressure 

 and length is the discontinuous curve CBAE, shown in 

 the diagram by a thickened line. 



The changes in the form of the collapsed cross-sections, 

 thus predicted, are clearly shown by experiments *, and the 

 diagram (fig. 3) throws an interesting light upon the well- 

 known formula of Fairbairn f , which makes the collapsing- 

 pressure inversely proportional to the length of the tube. 

 This result is precisely what the present investigation would 

 lead us to expect, if the experiments were restricted to short 

 tubes, and if the possibility of discontinuities at points such 

 as A and B in the curve of collapsing pressure were neglected ; 

 for the hyperbolic curve 



n 1 -n 2 = 464g 



is an almost perfect envelope of the discontinuous curve 

 CBAE down to the point at which the collapsing pressure 

 reaches a sensibly constant value. Fairbairn, by expressing 

 the whole range of his results in one formula, did neglect 

 ihe possibility of discontinuities, and it is obvious that his 

 experiments, since they show a steady fall of collapsing 

 pressure with increase of length, must have been restricted 

 to comparatively short tubes. 



December 13th, 1912, 



LXXI. On Conformed Representation from a Mechanical 

 Point of View. By Lord Rayleigh, O.M., F.R.S.t 



IN what is called conformal representation the co-ordinates 

 of one point x, y in a plane are connected with those of 

 the corresponding point f, n by the relation 



*+*y =/(£+*'*). C 1 ) 



where / denotes an arbitrary function. In this transforma- 

 tion angles remain unaltered, and corresponding infinitesimal 

 figures are similar, though not in general similarly situated. 

 If we attribute to f, rj values in arithmetical progression with 

 the same small common difference, the simple square net- 

 work is represented by two sets of curves crossing one 

 another at right angles so as to form what are ultimately 



* Of. W. C Unwin, ' Elements of Machine Design,' Part T. (1909) 

 p. 113, and A. P. Carman, University of Illinois Bulletin, vol. iii. No. 7. 

 t Phil. Trans. Roy. Soc. cxlviii. p. 389. 

 I Communicated by the Author. 



