1080 HON. LORD M'LAREN ON SYSTEMS OF 



x, y, z for a central plane, and it is evidently impossible to determine a contour-line of 

 the surface parallel to a central plane. Because if, for example, we make z = a, the 

 equation in x and y is thoroughly heterogeneous, containing in fact all the even terms of 

 the general equation of the 6th degree. 



But if the equation be presented for solution in the form (2), we can find values of 

 x and y in contour series. For we have then only to find a series of values of v\ v\ 

 (or x*+z 2 and y 2 + z 2 ); then, making z = a, we find from the series of values v ltl v 2}1 ; 

 Vi, 2 v 2 , 2 ; v 1)3 v 2 , 3 , &c, the coordinates 



Xi= Jv u l — a 2 ; x 2 = Jv!,l — a 2 ; x 3 = Jv^ — a 2 ; &c. 



y-i— Jv 2 ,l — cf; y 2 = Jv 2 ,i-a 2 ; y 3 = Jv 2 ,l — d 2 ; &c. 



Through such a series of points a contour-line of the surface, in the plane z = a, may 

 be traced. 



In the same manner contour-lines may be traced for other planes parallel to XY, 

 viz. z = b ; z = c, &c. 



Such contour-lines have been computed and traced for surfaces derived from the 

 curves (a) and (e). 



Plate VI. fig. 1, represents four contour-lines of the above surface (Eq. 2), with the 

 coefficient P = 1. The values of v x v 2 were taken from the preceding Tables (curve a, 6). 

 The maximum value of z was found to be, z ='7938 ; and the three inner contour-lines 

 were found by taking z successively equal to ^z , §z , and fz . The outermost contour- 

 line of this figure is the equatorial section of the surface, in the plane, z = 0, and is 

 identical with the curve of the Table, which is also figured in PI. I. 



It will be observed that as the circumference of the contour-lines decreases, the 

 Variation of curvature within the curve becomes less, the limiting form being evidently 

 circular. 



Plate VI. fig. 2, represents a series of contour-lines for the hyperbolic surface of two 

 sheets, derived from (e, 12) of the Tables by writing x 2 +z 2 for v\ and y 2 +z 2 for v\. In 

 this instance I have been less fortunate in the choice of contour-lines, because the lines 

 are not far enough apart to give a clear notion of the figure of the surface. The values 

 of z 2 , from which the computations were made, are - 003, "0125, and '0275, and the results 

 are shown in the figure. 



I may here observe that, while the preceding illustrations are confined to symmetrical 

 forms, it is apparent that if the analytical expressions were varied by merely altering the 

 coefficients of the terms, such a variation would only affect the symmetry of the curves, 

 and would not in general produce a curve of a different type. There is no difficulty in 

 forming any number of systems of unsymmetrical curves or contour-lines of surfaces, as 

 we have only to fix on any unsymmetrical homogeneous expression in v x v 2 , and to replace 

 these quantities by Jax 2 +z 2 and *]by l -\-z\ giving such values to z as may be desired ; 

 x and y are then found from v, v 2 . 



