for Multiply-Connected Regions of a Surface ^ 237 



a simple cut which is added to the boundary does nut divide; the 

 region, the anguh^r excess is reduced by 27r, and, since the con- 

 nectivity is reduced by unity, the etfective angular excess is 

 unaltered. If, on the other hand, the cut divides the region into 

 parts of connectivities i, j, not only is the sum of the actual angular 

 excesses of the boundaries of the parts the actual angular excess 

 of the original boundary, but, since /; is i +j — 1, the sum of i — 1 

 and J — 1 is ^' — 1 : the effective angular excess of the boundary of 

 the whole is the sum of the effective angular excesses of the 

 boundaries of the parts. Effective angular excess is therefore 

 additive in precisely the same way as the surface integral of a 

 single-valued function. If then Bonnet's theorem for a simply- 

 connected region is expressed in the form that the sum of the 

 integral curvature and the integral geodesic curvature is the 

 effective angular excess, the restriction on the connectivity is seen 

 at once to be superfluous. But to take this course implies a 

 previous acquaintance with the theory of connectivity, whereas it 

 is arguable that if Bonnet's theorem is used to establish the theory 

 of connectivity the extent to which there is an appeal to intuition 

 is materially reduced. 



