JVewtou and l*/ifl//ps on certain Transcendental Curves. lOT 



28. The resemblance of lig. 147 to a series of contour lines in sur- 

 veying, suggests a corresponding treatment of the equation. Let 



2:=sin// sin/y?y— asina- ^mn.r—b (19) 



be the equation of a surface, and let it be intersected by planes 

 parallel to the plane of .*-y, and we may obtain the groups of curves 

 described in Arts. 22-27. 



The surface of equation (19) may be described by continuous mo- 

 tion, as follows : Let sr=:siny sin my be a plane curve (figs. 1-37), and 

 let it move parallel to itself so that each point of it shall describe a 

 straight line parallel to the axis of .v. The curve shall then describe 

 a cylindrical surface whose equation is 



zz=.%mi/ •t^mtny. (20) 



Let z=z — am\x^\\\nx — b be the equation of a second plane curve, 

 and let this curve move parallel to the plane xz, in such a manner 

 that the axis of x of the curve shall always lie in the cylindrical sur- 

 face (20), The curve will describe by its motion the surface of 

 equation (19). 



The surface will consist of one contini;ous sheet lying between the 

 two parallel planes sr^it (l + ^-j-^'*), the positive numerical values of 

 a and h being here taken. 



29. By means of the two arbitrary constants, a and h, in equation 

 (1) the curve may be made to pass through any two points of the 

 plane. 



In a rectangle whose base is 2*7' ;r, and altitude IqTt, there are 

 '^{p-\-q){p' -\-q') possible positions of double points (Art. 11, k.) If 

 the curve passes through such a point it must have there two branches 

 real or imaginary. 



Hence we may assign to a and b such values that the curve will 

 have double points, in general, at any two of the ^{l^-\-q){l)'-{-q') 

 possible positions. 



ERRATUM m PLATE XVI. 



In figure 40, plate XVI, there is a series of ovals about one-half of the real double 

 points. There should be added to the curve, as represented, a like series of ovals 

 around each of the remaining real double points. 



