50 THE UNIVERSITY SCIENCE BULI^ETIN. 



closed circuits on F may be reduced to zero or to sums of mul- 

 tiples of four irreducible circuits. 



Having considered the elliptic case in detail and investigated 

 briefly a special hyper-elliptic function, we now proceed to 

 the most general hyper-elliptic function, w = R{z)f where 

 R{z) is of degree n. 



Forming the equation of the surface F in the usual manner, 

 there arises the equation F{x, y, u)^ 0, where F is of degree 

 2n in (x, y, u) . F(x, y,u)^0 may always be put in the form, 

 4^4 _ 4^25 _ 7^2 ^ 0, 



where S and T are polynomials in x and y of degree n. As has 

 been shown in the preceding considerations, R(z) may be 

 assumed to have only real roots. Hence the surface F is 

 symmetric to the XU and XY planes. The XU trace will con- 

 sist of the XX axis and a curve representing all real pairs 

 (w, z) satisfying the original equation. The latter curve will 

 consist of one or two infinite branches, according to whether 

 n is odd or even, and 2^ ovals. The XY trace will be a double 

 curve represented hy T = and consisting of the XX axis 

 and a curve represented by an equation of degree (n — 1). 

 This double curve represents all the real double points of the 

 surface F. 



The surface F is composed of two sheets which hang to- 

 gether everywhere along the XX axis except for values of x 

 which satisfy the equation u^ R(x) ,y ^0, and cut each 

 other along certain branches of the double curve T ^0. Cor- 

 responding to the p ovals there will be p holes in F. All closed 

 circuits on F may be reduced to sums of multiples of 2p irre- 

 ducible circuits. 



DOUBLE CURVES. 



The double curves of the surface F arise as the result of 

 projecting the surface $ from four space into three space, 

 the center of projection being at infinity. Whenever a pro- 

 jecting line cuts ^ in two places a double point occurs on F. If 

 the two points on <t> be real the double point on F will be a real 

 double point connected with the surface F, but if the two points 

 on $ be imaginary the resulting double point on F will be 

 isolated. This gives rise to two classes of double curves, one 

 being on the surface F and the other being related to the sur- 

 face but isolated from it. 



