ON THE 1'(J1N( AHF, 1 R aNSFOHM ATION 441 



of any point P and its image P, while ( and g are functions which by hypoth- 

 esis are continuous and single valued within the ring (Cc) and en its boun- 

 daries. The same is true of the quantity 



(2) Z = e — = (r, G) 



which measures in value and sign the angle POP. I shall call Z th« deviation 

 for the point P. The following are properties of this function which im- 

 mediately follow from the hypothesis. 



The deviation is positive for any -point of the inner contour (c), negative for 

 any point of the outer contour. (Hypothesis c.) 



On any ray OM there exists at least one point D for which the deviation is 

 zero. Such a point is shifted by the transformation radially only t. e. D and D 

 are collinear with 0. 



4- The locus of zero deviation. 



The locus of all points D within the ring for which the deviation vanishes 

 has for equation 



(3) z = (r, 9) = O 



I shall denote this locus by (D). The transformation exercises on this 

 locus a central effect shifting every point D on it along the ray OD. It follows, 

 therefore, that 



// E is a multiple point of order p on (D), E is a multiple point of the same 

 order on (D), and E and E are collinear with 0. 



If a ray 1 touches (D) in A it will also touch (D) in A, and the contact is 

 of the same order. 



If (D) possesses loithin (Cc) a cl sed branch (u) enclosed between two rays 

 1 and i' the image (u) is also closed and is contained in the same angle. 



All these properties are immediate consequences of the hypothesis and 

 definitions. 



6. The Principal Branch. 



Lemma A. The iocus of zero deviation has within the ring (Cc) at least one 

 closed branch (d) completely surrounding the inner boujidary. 



Indeed, if we regard (3) as the equation in semipolar co-ordinates of a 

 surface S, the cylinders parallel to Oz and built on (C) and (e), will meet S 

 in two curves (r) and (7) of which r is entirely below the plane n while y 

 is entirely above. The portion of the surface contained between the two 

 cylinders is continuous and single sheeted. S therefore, will be cut by n in 

 at least one closed branch completely surrounding (c). But -the complete 

 section of S by n is the locus (D), which proves the lemma. 



The branch (d) may have multiple points, _but if (E) be such a loop on 

 (d), the image (d) will possess a similar loop (E). The elimination of loops 

 on (d) will have, therefore, the effect of eliminating the loops on (d). It is, 

 therefore, legitimate to assume that (d), is a simple contour, as well as its 

 image (d). 



