emch: on a special class of connected surfaces. 157 



The study of complex figures resulting from any number of loup- 

 cuts drawn in either a unifacial or a bifacial surface would be the next 

 step; but all the problems of this sort can be solved by combinations 

 of the simple cases, and so may be left as exercises for the reader. 



Finally it is well to mention the fact that the well known theorem 

 of Gauss could not be employed in accordance with the treatment 

 herein given. Suppose the boundaries of the bifacial surfaces to be 

 closed curves, and x, y, z, the co-ordinates of any point in the one 

 curve and x', y', z', the co-ordinates of any point in the other, and 

 /\ the determinant 



X — x' y — y' z — z' | 



dx dy dz 



dx' dy' dz' 



The double integral extended over both lines, 



A 



ff 



■^m n. 



J -;(x-x')2 +(y-y')^ .{_(z-z')^^| 

 where /// is the number of twists, expresses the theorem of Gauss.* 



In order to aid in the practical calculation of the integral, one of 

 the curves can be brought by bending into a horizontal plane. 

 A point that describes the second boundary in the same direction 

 may pass a twist either by crossing the plane from above to below, or 

 from below to above. Thus there are two kinds of twists to be 

 distinguished in the solution of the integral; anil its numerical value 

 is 4"i; (a — b)n, in which a is the number of twists of the first kind 

 and /' is the number of the second kind. 2 is + 1 or — - > according 

 as a — /> is positive or negative. In the example discussed above a 

 distinction between the two kinds of twist was not necessary, and 

 therefore the numerical value of Gauss' integral is illusory and of no 

 avail in those cases. 



*0. Simonj-. Math. Ann:Uo!i. Vol. XXIV. p. 378. 



