234 Mr Neville, The Gauss-Bonnet Theorem 



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The Gfinss-Bonnet Tlieorem for Multiply -Connected Jler/ions of 

 a Surface. By Eric H. Neville, M.A., Trinity College. 



[Received 1 Dec. 1918: read 8 Feb. 1919.] 



Among the most delightful passages of differential geometry is 

 the use of Green's theorem to prove the relation discovered by 

 Bonnet between the integral curvature of a bounded region on 

 any bifacial surface and the integrated geodesic curvature of the 

 boundary. The fundamental equation is 



,ds+l\Kd^S= I'^^ds, 



.V 'as 



where the line integrals are taken round the whole boundary and 

 the surface integral over the region contained, Kg is the geodesic 

 curvature of the boundary, K the Gaussian curvature of the 

 surface, and f an angle to the direction of the boundary from the 

 direction of one of the curves of reference. Though there is no 

 allusion to curves of reference on the left of this equation, not 

 only do these curves appear explicitly on the right, but the use 

 of Green's theorem implies that there does exist some system of 

 curvilinear coordinates valid throughout the region and upon the 

 boundary, an assumption of which it is difficult to gauge the exact 

 force. The primary object of this note is to express Bonnet's theorem 

 in a form purely intrinsic. 



In the case of a simply-connected region not extending to 

 infinity, whose boundary has continuous curvature at every point, 

 the value of J(d^/ds)ds is 27r*. If the region is simply-connected 

 and does not extend to infinity, but the boundary is a curvilinear 

 polygon, formed of a finite number of arcs of continuous curvature, 

 the sum of the external angles must be added to the integral to 

 make the total of 27r ; in other words, j{d^/ds) ds is then the 

 amount by which the sum of the external angles falls short of 27r. 

 In the particular case of a curvilinear triangle, the amount by 

 which the sum of the three external angles fails short of 'Itt is the 

 amount by which the sum of the three internal angles exceeds tt, 

 and is called the angidar excess of the triangle. The name is 

 adopted to serve a wider purpose : whether a connected region of 

 a surface is bounded by a single closed curve or by a number of 



* See a paper by G. N. Watson, "A Problem of Analysis Situs"', Froc. Loud. 

 Math. Soc, ser. 2, vol. 15, p. 227 (1916). 



