SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1081 



16. Central Curves whose Equations are of the Form F a (x, y) n = ¥ 2 (x, y) n ~ p . 



The first function may be divisible by the second, without remainder, the equation 

 being then reducible to one of lower degree. This will generally be the case where the 

 equation consists of pairs of homologous terms, all of which have like signs, or all unlike 

 signs. 1 here suppose that the equation is not divisible. 



Confining our attention, as before, to symmetrical diametral equations, it is evident 



that such equations always contain at least one uncombined power of the variables, 



because, if the equation be given in composite terms, we can always divide out the lowest 



powers of x and y. When the equation after reduction consists of only two homogeneous 



parts its form is easily determined. Transforming to polar coordinates and dividing 



f p (cos 6, sin 0) 

 by the lowest power of r, we obtain an equation of the form r n ~ p = j^, 4—. — -A. The 



denominator of this fraction is formed from the terms of the highest homogeneous part, 

 and if its terms be all positive, r cannot become infinite ; but if the numerator be wholly 

 positive and the denominator contains positive and negative terms, there will be 

 certain values of d for which r is infinite, these being the same as were found for 

 the curves /?, e, £ (p. 1076). Again, if the numerator consists only of positive terms, 

 the curve cannot pass through the origin, as it necessarily does where some of the 

 terms in the numerator are positive and some are negative. If the terms of 

 the highest homogeneous part be all positive, and the terms of the lower degree be 

 partly positive and partly negative, the curve will be of the " foliated " type, consisting 

 of a series of loops symmetrically arranged about the centre-origin, and having no 

 inflexions except at the centre where the trace passes from one loop into another. 



More generally, for symmetrical expressions of any even degree, and any number of 

 pairs of homologous terms of even powers of the variables equated to zero ; which may 

 be written u n + u n _ 2 + . . . u 2 = : and are supposed to be reduced to their lowest terms, — 



(1) If any pair or pairs of homologous terms of the part u n have unlike signs, while 

 the terms of lower degree are all positive, or are all negative, then, by transforming to 

 polar coordinates and dividing by r", we find that r n = co is a solution of the equation 

 where u n — 0. The curve, therefore, consists of branches of infinite extent resembling 

 those already described under the character of contour-lines of surfaces formed from the 

 equations (e) and (£). 



(2) If u n consists entirely of positive terms, and if any pair or pairs of terms in the 

 parts of lower degree have unlike signs, and the other pairs are all positive, then the 

 curve consists of finite branches or loops passing through the centre. Because (l) the 

 radius-vector cannot become infinite, since u n consists of positive terms, and (2) when 

 cos = sin 6, all the negative terms are neutralised by the homologous positive terms, and 

 there remains a series "of positive terms equated to zero ; whence r = 0. In this case, 

 since cos 9 = sin 6 when r = 0, the tangents at the centre bisect the angles between the 

 axes of reference, and are secondary axes, and the centre is a point of inflexion. 



