Volume 1

Collected papers of R. A. Fisher / edited by J.H. Bennett.

  • Ronald Fisher
Date:
1971-1974
Catalogue details

Licence: Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)

Credit: Collected papers of R. A. Fisher / edited by J.H. Bennett. Source: Wellcome Collection.

    75/616 (page 67)
    In the case of lines whose direction is discontinuous, we must allow an inward curvature at each corner equal to the external angle; thus, for a geodesic triangle, the sum of the external angles is 2 K-jjGdS, whence that of the internal angles is 7r -f 11 G d S. It is important to observe that the geodesic curvature of any line on a surface, and therefore the Gaussian measure of curvature of any area, is unaltered by bending an inextensible surface; this is perhaps evident geometrically, but it can also be simply proved from our definition. After deformation, each point is represented by the same coordinates A. and ¡x as before, and for an inextensible surface r,- r 2 , and n will be the same in magnitude; we may there fore represent a small deformation by writing r, + doo xr, for r,, rj + dco x r 2 for r 2 ? n + d^xn for n; and the geodesic rale of curvature will be unaffected if n*r, xr u is invarible. Now for r M we put r a + d,xr, + d<wxr n , and for n x r, we have nxr, — r,n*dw + nr,*d, so n xr, • r n becomes nxr/r,, — rpr^n'dw + n*r u r 1 , dw + (nxr,)*(do) x r n ) + (nxr,' • dco, x r, =nxr,*r 11 + r/hi'dco,. But r l2 becomes r ]2 +d&>. 2 xr, + dct)xr 12 , or r 12 +dw, x r 2 +dto x r,,, therefore daq x r, = day x r.,, and so dco, and dco., must lie in the tangent plane; hence n* dco, = 0, so that the geodesic rate of curvature is unaltere l.