for Multiply-Connected Regions of a Surface 235 



curves, the amount by which the sum of all the external angles 

 of the boundary falls short of 27r is called the angular excess of the 

 boundary. 



Whatever the number of curves forming the boundary of a 

 region, the addition to the boundary of a simple cut, joining a 

 point of the boundary either to a point of the cut or to a point of 

 the boundary and described once in each direction, increases the 

 sum of the external angles by 27r. If the cut divides the region 

 into two parts, the angular excess of each part is the amount by 

 which the sum of the external angles of that part tails short of 27r, 

 and therefore the sum of the two angular excesses is the amount 

 by which the sum of the external angles of the composite boundary 

 falls short of 47r; this, being as we have just seen the amount by 

 which the sum of the external angles of the original boundary falls 

 short of 27r, is the angular excess of the original boundary. If on 

 the other hand the cut leaves the region undivided, there is an 

 actual decrease of 27r in the excess. It follows that if by a 

 succession of n simple cuts the region is divided into m distinct 

 parts, the sum of the angular excesses of the boundaries of the 

 parts is less than the angular excess of the original boundary by 

 2 (?i — in + 1) IT. Suppose now that each of these parts is simpl}^- 

 connected and that there are no singular points of the surface ni 

 the original region or upon its boundary. Then since Bonnet's 

 theorem in its simplest form is applicable to each of the parts, 

 addition of the sum of the integral curvatures of the parts to the 

 sum of the integral geodesic curvatures of the boundaries of these 

 parts gives the sum of the angular excesses of the individual 

 boundaries. But the sum of the integral curvatures of the parts 

 is the integral curvature of the original region, and the sum of the 

 integral geodesic curvatures of the boundaries of the parts is the 

 integral geodesic curvature of the original boundary, since an arc 

 described once in each direction adds nothing to JKgds. Hence 

 the sum of the integral geodesic curvature of the original boundary 

 and the integral curvature of the bounded region is less than the 

 angular excess of the original boundary by 2 {n — ni + 1) tt. This 

 result affords a proof that if only the dissection has reached a stage 

 at which every part is simply-connected, the difference n — m is 

 independent alike of the form of the cuts and of their number. 

 Since a simply-connected region is divided by one cut into two 

 pieces, the integer used to measure connectivity is not n — m but 

 n — in -h 2, and Bonnet's theorem in its most general form asserts 

 that 



If a bounded bifacial region of any surface has finite con- 

 nectivity k and neither extends to infinity nor includes tvithin it or 

 upon its boundary any singularities of the surface, the sum of the 

 integral geodesic curvature of the boundary and the integral curva- 



