126 Mr Burnside, On a paper relating [Nov. 24, 



demonstrating the relative movements of the earth, sun, moon ana 

 stars is more particularly intended to illustrate and explain, ohe 

 astronomical observations by means of which the position of an 

 observer is determined on the earth's surface, e.g. meridian and 

 ex-meridian altitudes of the sun or a star whether the latter be 

 circumpolar or not. It of course affords a clear explanation of the 

 more important terms used in navigational astronomy, serves also 

 to illustrate the cause of the varying length of day and night at 

 different seasons of the year, the phases of the moon and their 

 relation to the tides, and affords a rough demonstration of the 

 course of solar eclipses. 



(5) Note on a paper relating to the Theory of Functions. By 

 W. Burnside, M. A., Pembroke College. 



In a paper " on the geometrical interpretation of the singular 

 points of equipotential curves " printed in Vol. VI. of the Society's 

 Proceedings, Mr Brill has stated some properties of algebraical 

 equipotential curves which are of very doubtful accuracy. 



The point of view taken seems to be this : that the several 

 members of an equipotential family of curves do not generally 

 meet in real points at all, and that when they do the points of 

 intersection of the consecutive curves of the family are the branch 

 points of the function which gives rise to the family. 



It is not explicitly stated, though it seems to be implied, that 

 these points, viz. the branch points, are the only real points of 

 intersection of such a family. Several of the results obtained in 

 the paper in question depend on the accuracy of the above state- 

 ments, so that it is perhaps worth while to examine them in 

 some detail. 



For this purpose I limit myself to the case in which not only 

 the curves themselves are algebraical, but also the function which 

 gives rise to them. I also consider, except in the last paragraph, 

 real points on the curves and real points only. 



Suppose that f{z, w) = is an equation of the nth degree in 

 w. If oc + iy and u + iv are written for z and w, two equations 

 each connecting x, y, u, v will result from f— 0, and by eliminating 

 u and v alternately between these the equations of two systems of 

 algebraical curves will be obtained of the forms 



fx 0. y> u ) = °> / 2 0> y> v ) = °- 



The clearest conception of the intersections (real) of these curves 

 may be obtained in the following manner. 



The function w of z, which as defined by the above equation is 

 w-valued, can be represented as a one-valued function on the n- 

 sheeted Riemann's surface belonging to the equation /=0; and 



