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Published online by Cambridge University Press: 24 October 2008
Beltrami's discovery that corresponding to any ruled surface there is another applicable on it having corresponding generators parallel but with the parameter of distribution of opposite signs, has brought into prominence the distinction between the applicability of two surfaces and the continuous deformation of one into the other. It is pointed out that since the parameter of distribution differs in sign, the tangent plane rotates in opposite directions as the point of contact proceeds along the two corresponding generators on the respective surfaces. One surface cannot, therefore, be deformed into the other. It is proposed in the present paper to investigate the distinction between applicability and deformability of two surfaces in general, and to connect this isolated fact with the general theory of deformation.
* Ann. di Mat., t. 7 (1865), pp. 139–150, or Forsyth, Differential Geometry, p. 387.
† See, for instance, Forsyth, loc. cit., p. 376.
* These commonplaces of Differential Geometry have been reproduced to emphasize their analytical bearing on the problem of deformation.
* Forsyth, loc. cit., p. 50.
* Forsyth, loc. cit., p. 388.