374 Prof. F. L. 0. Wadsworth on 



is unity and od is constant and equal to — - — , the sides ob, 



de, of any pair represent coincident values of x and y in the 

 corresponding hyperbola. These coincident values laid off 

 along their respective axes give a series of points on the 

 curve from which it is easy to trace the curve itself. In 

 practice, the whole operation may be rapidly and easily per- 

 formed by means of a T-square and a single triangle of which 

 one angle is equal to the angle yox between the axes *. First 

 draw the line mn (fig. 2) parallel to and at unit distance 

 from the axis of x, and the line dg parallel to the axis of y 



a? + b 2 

 and at a distance from it equal to — - — . Then to the points 



c d c" on the first line draw the lines oc oc 1 oc" &c. by the aid 

 of the edge of the T-square or an auxiliary ruler, and the 

 lines cb, c'b' ', c n b" by means of the triangle or bevel-gauge. 

 Project the points of intersection e, e' ' , e" of the first set of 

 lines with the line dg upon the second set of lines, giving us 

 the points s, s', s" on the required hyperbola. This method 

 is particularly rapid and convenient in plotting rectangular 

 hyperbolas on cross-section paper, the only instrument then 

 necessary being a rule to draw the radial lines oc, oc', &c. 



If desired, an instrument can easily be constructed on these 

 lines to trace the curve mechanically, but generally the 

 graphical process is more rapid. A whole series of contours 

 to the thermodynamic surface (pv = const.) can be drawn by 

 this process in a very few minutes, the same set of lines oc, 

 oc', oc", and cb, c'b', c"b", answering for all the curves. 



2nd method. — Make the vertices of the two similar triangles 

 coincide at c instead of as before, and make ab = oc = x, and 

 cd=y (fig. 3). Take a point a on the axis of y at unit dis- 

 tance from the origin and draw from it the lines ac, ac' , ac" 

 to points on the axis of x, and the lines be, b'c', b"c" parallel 

 to the y axis as before. Mark off a distance equal to 



fd = — j — on the edge of a triangle (or bevel-gauge), Q, of 



which the angle at d is equal to the angle yox, and slide this 

 triangle along each of the lines be, b'c', b"c", &c, until the 

 point / intersects the corresponding line ac, ac', or ac". The 

 points d, d', d" will then evidently be points on the hyperbola. 

 In practice it is not necessary to draw the lines be, b'c', &c. 

 at all, it suffices to place a T-square whose blade is parallel 

 to the axis of y, so that its edge passes through any of the 



* It is convenient to use for this purpose an ordinary steel bevel- 

 gauge the two blades of which may be adjusted to the required angle. 



