1084 HON. LORD M'LAREN ON SYSTEMS OF 



Equation (2a) represents the same curve as (2) transformed to axes inclined to the 

 former at the angle 22° "5, and having the axes tangents at the centre origin. Values of 

 r are impossible for each alternate arc of 45°. The curve consists of four equal, similar, 

 and symmetrical loops or foliations, and the centre is a point of inflexion for the four 

 intersecting lines (fig. 3). Fig. (4) is a linear projection of the same curve. 



In this and the preceding figure the bitangents are seen to be parallel to the conju- 

 gate axes. 



The same construction is evident in fig. 4, which is a projection of the last- 

 mentioned curve. 



Equation (3). 



0= 



0° 



10° 



15° 



20° 



22°-5 



23° 



i 

 23°10 



r = 



1 



103 



115 



1-37 



1-68 



326 



00 



Plate III. fig. 5, represents this curve, which consists of four hyperbolic branches 

 without inflexions. As these branches do not pass through the centre, although u 2 

 contains a negative term, it is evident that the equation is reducible to one of lower 

 degree with an arbitrary term. Accordingly, by division we find for the reduced form of 

 equation (3), — cc 4 + y 4 —5x 2 y 2 = A 6 , or say, = 1. 



17. Contour-lines of Surfaces derived from Central Curves passing through 



their Centres. 



Having already explained the mode of derivation of such lines, it is here only 

 necessary to describe the illustrative figures (PL VI. figs. 3 and 4). 



Figure 3 represents a surface with a central core or axis, being the axis of Z. It is 

 formed from equation (1), above, by taking r 2 = r /2 +z 2 . z =l is a maximum, and the 

 four contour-lines are sections of the derived surface in the planes, z = 0, z = *65, z=75, 

 and z='9. 



Figure 4 is formed from Equation 2 (above) by transforming to x-and-y coordinates, 

 and then taking x 2 = x 2 +z /2 ; y 2 = y' 2 +z 2 . z =l is a maximum, and the contour-lines are 

 for sections of the derived surface in the planes, z = 0, z='l, and z=*14. 



In this surface the pear-shaped figures are only united at the cusps, which are also 

 points of inflexion, and the sections consist of detached loops. The diagram makes clear 

 what is the kind of variation of an equation by which a continuous looped curve may 

 break up into detached loops or ovoids. We see that these only become continuous 

 through the disappearance of the quantity z, and by the equation becoming a homo- 

 geneous function of x and y of the form, 



F B (a!,y)/F B .>,y)=l. 



It will be understood that these contour-lines are all traced from a sufficient number 



