

SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1073 



(3) Another series of critical forms is the series where the coefficient P=l. By 

 making P = 1 in the four equations (a). (/3), (e), and (£) we obtain curves which may also 

 be obtained by the multiplication of the factors (x 2 +y 2 ), (x 2 — y 2 ) with (x i + y i ) and 

 (x i —y i ). The forms (ft) and (£) are also obtained by the multiplication of (x 2 =hy 2 ) 2 with 

 (x 2 =p y 2 ). 



(4) A fourth series of critical forms are those which correspond with the polar 

 equations of sines and cosines of multiple arcs. These will be noticed in their order. 



(14c). Form and Variations of the Equiaxial Curves (a) and (ft). 



It is desirable to give a name to the variation of the curve consequent on the variation 

 of the single coefficient P. 



In equations of the 2nd degree the term "eccentricity" has relation to the 

 variation due to projection, which is the only kind of variation of which these curves 

 admit. 



But in curves of the higher degrees, where we consider only those characteristics that 

 are unaltered by projection, this kind of eccentricity is not considered at all. Hence, 

 without ambiguity, I may make use of the term Quadrantal Eccentricity to denote the 

 variation within each quadrant in the magnitude or direction of the secondary axes due 

 to the variation of P, while the principal parameters remain unaltered. 



The quadrants referred to are of course those which are marked out by rectangular 

 reference lines, coinciding with the equal principal diameters or parameters of the 

 curve. 



Length of a Secondary Diameter in terms of the principal Diameter and P. — The 

 variation of P in the oval of fourfold symmetry, has no effect on the direction of the 

 secondary diameters, but only alters the ratio of their length to that of the primary. 

 It is convenient always to put the length of a principal diameter =1, which is then also 

 the value of the arbitrary term. The ratio of the secondary diameter to the primary 

 may be denoted by T, which is also numerically the length of the former. In the oval 

 forms of (a) and (/3) we then find for the length of the secondary diameters, by putting 

 x = y ; r 2 = 2x 2 , the relation, 



2P + 2-i-^- • 1 - P+1 - P-l-l 



In curves of the hyperbolic type where the variation of P affects the direction of the 

 asymptotes, the quadrantal eccentricity might be measured by the tangent of their 

 inclination to the principal axis ; but this relation has not been fully investigated. 



Examples of the Curves, (a) and (/?). — The following equiaxial curves of the forms 

 (a) and (/3) have been computed and traced. The number in the first column is a 

 reference number corresponding with that in the second table ; P and T are as above ; 



