A b γ z fig b value of z and then following the

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( a ) ( b ) Γ Z Fig. B. value of Z , and then following the continuous variation of this value as Z moves along Γ. We shall, in the argument which follows, use the phrase * It is here that we assume that Γ does not pass through the origin. 545
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APPENDIX I 546 ‘the amplitude of Z ’ and the formula am Z to denote the particular value of the amplitude of Z thus selected. Thus am Z denotes a one-valued and continuous function of X and Y , the real and imaginary parts of Z . When Z , after describing Γ, returns to its original position, its ampli- tude may be the same as before, as will certainly be the case if Γ does not enclose the origin, like path ( a ) in Fig. B , or it may differ from its original value by any multiple of 2 π . Thus if its path is like ( b ) in Fig. B , winding once round the origin in the positive direction, then its amplitude will have increased by 2 π . These remarks apply, not merely to Γ, but to any closed contour in the Z -plane which does not pass through the origin. Associated with any such contour there is a number which we may call ‘the increment of am Z when Z describes the contour’, a number independent of the initial choice of a particular value of the amplitude of Z . We shall now prove that if the amplitude of Z is not the same when Z returns to its original position, then the path of z must contain inside or on it at least one point at which Z = 0 . We can divide γ into a number of smaller contours by drawing parallels to the axes at a distance δ 1 from one another, as in Fig. C . * If there is, on the boundary of any one of these contours, a point at which Z = 0, what we wish to prove is already established. We may therefore suppose that this is not the case. Then the increment of am Z , when z describes γ , is equal to the sum of all the increments of am Z obtained by supposing z to describe each of these smaller contours separately in the same sense as γ . For if z describes each of the smaller contours in turn, in the same sense, it will ultimately (see Fig. D ) have described the boundary of γ once, and each part of each of the dividing parallels twice and in opposite directions. Thus PQ will have been described twice, once from P to Q and once from Q to P . As z moves from P to Q , am Z varies continuously, since Z does not pass through the origin; and if the increment of am Z is in this case θ , then its increment when z moves from Q to P is - θ ; so that, when we add up the increments of am Z due to the description of the various parts * There is no difficulty in giving a definite rule for the construction of these parallels: the most obvious course is to draw all the lines x = 1 , y = 1 , where k is an integer positive or negative.
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APPENDIX I 547 γ z Fig. C. P Q Fig. D. of the smaller contours, all cancel one another, save the increments due to the description of parts of γ itself.
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