SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1091 



complete square, or a complete binomial expansion of px + qy. This would seem to be 

 the proper criterion by which an equation not referred to principal axes may be known to 

 be the expression of a parabolic curve. 



If the equation consists entirely of terms of even powers, there are two parabolas, one 

 on each side of the axis of X, which may in a sense be considered to be two branches 

 of a central curve, as in the following easy example : — 



y= o 



i 



2 



3 



4 



5 



6 



x= 



1057 



2379 



4-027 



5-981 



8-206 



10-670 . 





21. Biradial Coordinates. 



The homogeneous equations hitherto treated have been solved for loci determined by 

 Cartesian coordinates. The same equations and the same series of values as are above 

 found may be represented graphically under different coordinate systems, and so as to 

 produce curves differing widely in form and geometric properties from the curves of the 

 x-and-y system. 



If to avoid ambiguity we call the coordinates for which values were found v u v 2) these 

 may represent radii, angles, or trigonometrical quantities instead of lines drawn to 

 coordinate axes ; and the equations may be equations in r x and r. 2 ; X and 0. 2 ; r and x ; 

 or r and % (where r is the radius vector from a pole, and z is a perpendicular on a 

 directrix). This last system again may be immediately transformed into trilinear 

 coordinates by substituting x 2 +y 2 for r 2 . The homogeneous equation in sinflj and 

 sin 2 is evidently identical with the homogeneous equation of corresponding terms in 

 r 2 and r x ; because in the variable triangle composed of the two radii and the line joining 

 the foci the sides are proportional to the sines of the opposite angles. 



As an illustration of what may be done in a new direction with the homogeneous 

 equations already examined, the following chapter on a class of Biradials has been 

 written : — 



The radial coordinates, from foci F l5 F 2 , are denoted by ryr 2 ; and the distance 

 Fi F 2 by 2c. 



The equations here considered are of the form 



r?dbr| = A? (1); r»/A w ± »■?//*" = c 1l Jv n = A" . . (2), 



where the index n is an even number. 



I have not been able to come to a clear conclusion regarding biradial equations of 

 uneven degrees. On the one hand, if we seek to transform these to rectangular or 

 ordinary polar coordinates, it is necessary to square the equation twice to remove the 



VOL. XXXV. PART IV. (NO. 23). 8 C 



