1092 HON. LORD M'LAREN ON SYSTEMS OF 



radical, so that a biradial equation of the 3rd degree corresponds to an equation of 

 the 12th degree in rectangular Cartesian coordinates; while a biradial of the 4th degree 

 can be transformed into a quadric in Cartesian coordinates. On the other hand, when the 

 biradial curves of the 3rd, 4th, 5th, and 6th degrees are traced, they are found to be in 

 series ; and in this case, apparently, the degree of the Cartesian equation (contrary to the 

 general understanding of mathematicians on this subject) is not a criterion of the true 

 order of the curve. 



Plate V. fig. 8, represents this series of curves, that of the 3rd degree being the 

 nearest to the centre. The dotted curve is the biradial curve of the fractional degree 

 7/2. Each curve is laid down from nine computed points for the quadrant. 



Equation (l) is the Oval of double symmetry. Its equation in ordinary polar 

 coordinates reduces to a simple and easily remembered expression. Taking 0, the centre 

 of the oval for the origin of Cartesian and polar coordinates (x, y, R, 0) ; OFj = OF 2 = c. 

 For a point P in the curve, we have from the triangles OFjP, OF,P, 



r \ = OP- + OF 2 - 20Fj.OP.cos F x OP = E 2 + c 2 - 2cR cos 6 ; 

 ri = OP 2 + OF 2 . + 20F 2 .OP.cos F,OP = E 2 + r + 2cE cos . 



In the expansions of r\, 7% the uneven terms of R cos disappear. Thus by trans- 

 formation r\ + r\ = A 2 becomes 



( R 2 + v 1 - 2cE.cos 6) + (E 2 + c 2 + 2cE.cos 6) = A 2 , 

 R 2 = (A 2 - 2c 2 )/2 , the circle. 



For the radial equation of the 4th degree we have 



r\ + r\ = A ; (R 2 + c 2 - 2cR.eos 0) 2 + (R 2 + c 2 + 2cE.cos Of = A 4 

 (E 2 + c 2 ) 2 + 4c 2 R 2 cos 2 = A 4 /2 ; 



(x 2 + y 2 + c 2 ) + (2cxf = A 4 /2 .... (3). 



For the radial equation of the 6th degree we have 



r \ + r \ = A° ; (R 2 + c 2 - 2cE cos Of + (R 2 + c 2 + 2cR,cos Of = A 6 , 



(R 2 + c 2 ) 8 + (R 2 + c 2 )(2cR cos Of = A°/2 ; 

 {x^y 2 + c 2 f + {.,?+ f + c 2 ){2cxf = A G j2 . . . . (4). 



Cognate polar equations may be formed in the same way for radial equations of any 

 even degree, whence the equations in x and y may be written out. The equations are 

 homogeneous functions of the composite quantities (x 2 + y 2 + c 2 ) and lex. 



If we write v t v 2 for these expressions, and solve the homogeneous equation in v^ 

 for any point, we may then find x = v 2 /2c, and y= Jvx—x 1 —^. 



