1094 HON. LORD M'LAREN ON SYSTEMS OF 



geneous biradial curves may be resolved into functions of curves of the second degree 

 with variable elements. 



Hitherto I have considered the radial curve as the geometrical expression of a homo- 

 geneous radial equation of the simple form r? + 1 \ = A. It will now be shown that all 

 symmetrical homogeneous equations in r x r. 2 of the same degree are identical curves, the 

 eccentricity being dependent only on the choice of foci, or, which is the same thing, on 

 the ratio of c to A. This identity is proved by transformation to polar coordinates. 

 For this purpose, let r?.r£~ p and rl~ p .rf, be any pair of homologous terms of the 

 symmetrical radial equation. If definite values be given to the indices n and p, and the 

 transformation to polar coordinates be effected by the formula r 12 = R 2 c 2 ± 2Rc. cos 0, it 

 will be found that, in the addition of the transformed terms, all the terms of uneven 

 powers disappear, and that the resulting polar expression is identical with that obtained 

 from the sum of the terms r\ and r%. Thus, from each pair of homologous terms we 

 have the same polar expression multiplied by a coefficient, and the transformed homo- 

 geneous symmetrical radial equation has the form already found for the equation of three 

 terms, with a new value of A. 



Thus, if we transform the equation r\ + Vr\r\ + Vr\r\ + 7% = A G to polar coordinates, we 

 obtain from the two extreme terms the terms of the left side of equation (4), which may 

 be denoted by F^n) ; and from the mean terms we obtain P x F(r 1 ? , 2 ). The polar 

 equation then is of the form (4) with the right-hand term divided by (P + l .), or 



as 

 (R 2 + c 2 )s + (R 2 + c 2 )(2cR cos 6f = 



2P+2 



The proposition that the same curve or trace may be obtained from different homo- 

 geneous equations of the same degree is illustrated by fig. (1). This figure, as is the 

 case with all the illustrations, is drawn by tracing the curve through a series of com- 

 puted points laid down on diagram paper, never less than nine points for a complete phase. 

 This figure, when referred to the two marked exterior foci, satisfies the equation 

 r\ + r\r\ + r\r\ + r\ = 1 ; and when referred to the two marked interior foci it satisfies the 

 equation r\ + r\= 1. The computed values of r x and r 2 are those of the table, p. 1074, for 

 the equations of these forms in x and y. 



If the homogeneous radial equation contains a middle term of even powers of i\r 2 , its 

 equivalent in polar coordinates differs only from the expression found for a pair of 

 homologous terms in having the negative sign prefixed to all the terms containing cos 2 0. 

 I must, therefore, qualify the statement of the preceding paragraph by adding that, in 

 the case of homogeneous equations of the 4th and 8th degrees (and generally where the 

 index is divisible by 4), there are apparently two forms, one without and the other with 

 a middle term. But it docs not appear that this variation can have any other effect than 

 that of varying the coefficients of the terms multiplied by cosine 2 ^. The examples 

 which I have worked out are confined to equations of the 6th degree, in which, of course, 

 there is no middle term. 



