236 Mr Neville, The Gauss-Bonnet Theorem 



I 



ture of the region bounded is less than tlie angular excess of the 

 boundary by 2(^' — 1) ir. 



In other words, the sum of the two integrals and the external 

 angles of the boundary is 2 (2 — k) -k. 



Gauss' famous theorem on the integral curvature of a geodesic 

 triangle, which may be regarded either as the simplest case or as 

 the ultimate basis of Bonnet's theorem, is in no less need of modi- 

 fication if the region contemplated is multiply-connected. 



If a geodesic triangle on any surface has internal angles A, B,C 

 and connectivity k, and if the surface is regidar throughout the 

 triangle and on its perimeter, the integral curvature of the triangle 

 is A + B-\-G-{'2.k-l)7r. 



The application to the whole of a surface which, like a sphere 

 and an anchor- ring, does not extend to infinity, but has no 

 boundary, is interesting. A simple closed curve can always be 

 drawn to divide such a surface into two distinct parts, and since 

 its direction as the boundary of one part is opposite to its direction 

 as the boundary of the other part, the sum of the external angles of 

 the two boundaries is zero, and so also is the sum of their integral 

 geodesic curvatures. It follows from Bonnet's theorem that, if 

 there are no singular points on the surface and the connectivities 

 of the two parts are i, j, the integral curvature of the complete 

 surface is 2 (4 — i —j) it. Hence i +j is constant ; in order that a 

 surface which, like a sphere, is cut by any simple closed curve into 

 two simply-connected parts may be described as of unit con- 

 nectivity, the connectivity is measured by the integer i+j—1, 

 and 



If the connectivity of a bifacial surface which has no boundary 

 and no singular points and does not extend to infinity is k, the 

 integral curvature of the surface is 2 (3 — k) ir. 



A striking deduction made by Darboux from Bonnet's theorem 

 may be mentioned here. If on a complete surface there is any 

 family of curves such that the surface can be divided into a finite 

 number of parts throughout each of which this family provides 

 one set of curves of reference, the angle ^ of our first paragraph 

 can be measured from the curve belonging to this family, and 

 J{d^/ds)ds taken once in each direction over every part of an 

 imposed boundary is necessarily zero. Hence 



For there to exist on an unbounded bifacial surface, which does 

 not extend to infinity and is everywhere regidar, afainily of curves 

 which covers the surface and is wholly withoid singularities, the 

 surface must have integral curvature zero and must therefore be 

 triply-connected. 



In conclusion the subject may be presented in another form. 

 Let the angular excess of the boundary of a region of connectivity 

 k reduced by 2(A;- l)7r be called the effective angular excess. If 



