310 G. H. KNIBBS. 



have, consistently with (33) to (35a), and denoting the quan- 

 tities belonging to w by accents, p = ^p, 6' — \Q or \Q-\-tc, that is 

 to say, two numerically equal but oppositely directed values of w; 

 which may be regarded, not as separate functions, but as th« 

 branches of a — in this case, two-valued — function. For example, 

 if z describe a closed path passing through neither nor oo, without 

 going round the origin, as in Curve 1 Fig. 31, 6' will have (in the 

 instance mentioned) only half the range of the limit lines therein 

 shewn, i.e. half the amplitude of 0. Tf z make one circuit, 0' will 

 range through tt only; but if z make two complete circuits, 0' will 

 make a complete circuit 27r: or more generally — in this particular 

 case — for an even number of circuits by z, the w curves will be 

 closed, for an odd number they will not. Thus on sheets, lying 

 indefinitely near one another but distinguishable, two series of 

 values can be represented by a contiruous curve, provided that 

 the sheets are so constructed that one 3an pass from one into the 

 other, as indicated in Fig 32 (a). 1 In this figure, P is infinites- 

 imally close to P' but on the lower sheet; this passes pseudo-con- 

 tinuously at the cut or branch-line' 2, to oo, (at which the sheets 

 are joined) into the upper sheet. The nature of the junction is 

 indicated in Fig. 32 (d), except that the distance between the 

 lines is infinitesimal. 3 The sheets are joined at no other place 

 than the branch line, not necessarily a straight line. 



Various branch-points, 4, such as A and B, Fig. 32 (b), will be 

 required according to the form of the function to be represented, 

 e.g. : — w= vUz~a)(z — b)i would require a two sheeted-surface 

 with a branch line, as in the figure, and a four- valued function, a 



1 This is necessary since we can pass continuously from + z to - z. 



2 Called also branch-section or crossing -line. (Verzweigungsschnitt — ligne 

 de passage). 



3 It is obvious that the sheets are not parallel planes, and in passing 

 from sheet to sheet we must move suddenly at right angles even if only 

 an infinitesimal distance. The continuity is obviously only pseudo- 

 continuity. 



4 Verzweignngspunkt oder Windungspunkt. The surface is known 

 sometimes as a winding surface, and the branch-point of m sheets as a 

 winding point of the (m - l)th order. 



