By B. O. Peirce
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Additional info for Elements of the Theory of the Newtonian Potential Function (1902)
MULTIFORM FUNCTIONS. 47 round each branch-point and let the variable describe the same in succession, then this path can be deformed, without line passing over one of the branchpoints, into a closed line which, starting from branch-points. a and z , encloses all the where two branch- (Fig. ) lines round the individual branchpoints most simply, by describing round each one a small circle, and connecting each of these circles with must then be described 10. We twice, going will next illustrate by some examples, and at the % by a and coming.
Since in this case x is constant and can be easily obtained, by substituting the value of a? in the equations equal to 1 ? 2 z=z of y and v 2xy and eliminating y\ thereby we obtain the equation v 2 4(1 u) between the co-ordinates u and # of the point w, which shows that the locus of w is a parabola, = u = with its vertex at eter, the 1, ordinate its at focus at the focus, o, is and tangents at the points of intersection respond to c and d, it is c which the paramBy examining the and d which cor- of 2.
1. If we let the variable start In this z = is a branch-point. from the point # = 1 and describe the circumference of a circle round the origin, this is a closed line which encloses the branch-point. If the function w = V# start from the point 2 = 1 with the value w = -+- 1, and if we put = r (cos + i sin <), = 0. z = 1, r = 1 and z 4> If z next describe then at the point 4> of increasing in the of circle direction the the circumference increases and to constant and r remains 1, equal angles, If therefore the variable return to the point to 2 TT.