SEMINAR NOTES 397
of each and all may be made obvious, and the direction indicated in which any unknown value is to be looked for; just as the discovery of the element argon was predicted from Mendelejeff's chart by the law of the periodic functions of atomic weights.
Precisely this task (/. e., the representation) was undertaken by Rie- mann and the solution found in his conception of the ^-sheeted surface."
"Let w=^V 2. To one z in the Argand diagram correspond two values of w, whereas we wish to make two points correspond to two values of w. To these two points the same complex variable z should be attached. Instead of a single 2-plane we take two indefinitely thin sheets, one of which lies immediately below the other. To the one value of 2 correspond two places in these sheets, one vertically below the other. Every place in the upper or lower sheet has one, and only one, of the two values V 2 or — 1 z permanently attached to it. If for a given z, V^z be attached to the upper sheet, then —12 must be attached to the lower sheet at the point 2. When 2=0 or 03 , the values of w are equal, and only one place is needed to represent them ; hence we regard the sheets as hanging together at o and co . . . . If in the Riemann representation the path start from a place in the upper sheet to which the value 1 2 is attached, it must lead to that place in the lower sheet which lies vertically below the initial point. It follows that we must give up the idea of having only one 7i'-value attached to each place of the two sheets, or else make a connection or bridge between the sheets over which every path which goes once around the origin must necessarily pass."'
'For the formation, etc., see Harkness and Morley, Treatise on the Theory oj Functions, extracts from which are given below.
^ Directions for making model of a Riemann surface for a three-valued function : A difficulty arises, first, because the sheets of the surface interpenetrate, and, in the second place, because frequently at branch-points several sheets, which do not lie one immedi ately below another, must be supposed to be connected. For the purpose of illustration it is only necessary to be able to follow certain lines in their course through the differ- ent sheets of the surface. This may be done as follows : First cut in the three sheets of paper plaf ed one above another, which are to represent the surface, the branch-cuts, and then only at those places where a line is to pass over a branch-cut from one sheet into another join the respective sheets by pasting on strips of paper. Then we can always contrive that, when the line is to return to the first sheet, from which it started, we have the necessary space left for the fastening of the strip of paper by means of which the return passages is effected. By these attached strips union of the separate sheets into one connected surface is accomplished ; and it is then no longer necessary to connect the sheets with one another at the branch-points.
