1064 HON. LORD M'LAREN ON SYSTEMS OF 



(6) The inclination of the asymptotes of a heterogeneous central curve is always the 

 same as the inclination of the asymptotes of the curve represented by its highest homo- 

 geneous part. Because, if we transform to polar coordinates, and divide the polar 

 equation by r n , the resulting equation is of the form — 



F (cos 6, sin 6f + F^cos 6, sin 8) n - 2 .- 2 + F 2 (cos d, sin 6f ~ i . \ . . . = \ . 



Now, r can only become infinite when F (cos 9, sin 6) n = 0. 



But this is also the condition for r becoming infinite when the equation is reduced 

 to its highest homogeneous part. 



It follows that for all parallel sections of a central surface the inclination of the 

 asymptotes to the axis of symmetry of the section is the same, and it is evident that all 

 such asymptotes lie in two intersecting planes. 



(7) In the case of the homogeneous central equations with an arbitrary term, it is 

 evident that the curve cannot pass through the centre. 



11. Transformation to Secondary Axes — Rule of Signs. 



A symmetrical equation is evidently equiaxial ; that is, the intercepts on the axes of 

 reference are equal. 



If a diametral symmetrical equation be transformed to secondary axes (bisecting the 

 angles between the primary), the secondary axes are also diametral and symmetrical, 

 and the curve consists of eight equal and similar segments. This might be inferred from 

 general considerations as to symmetry, but it is desirable to prove it analytically. It 

 may be here convenient to transcribe certain known formulas of transformation of axes 

 (with unchanged origin) of which I am to make use. If 6 be the inclination of X to x — 



X = xcos$ — i/sin 6 Y = xsmd + y cos0 . . . . (1) 



X = (cc-2/)cos0 Y = (x + y)sxaQ (2) 



X = (x-y)jl Y=(x+y)Jh (3) 



(1) Is the formula for transformation in the same plane from any system of rectan- 

 gular coordinates to any other rectangular system. 



(2) Is the formula for transformation to symmetrical axes ; i.e., axes equally inclined 

 to the original rectangular axes. 



(3) Is the formula for transformation to axes which are at once symmetrical and 

 rectangular, and which accordingly bisect the angles between the original rectangular 

 axes ; whence, cos 6 = sin 9 = J\ . 



In order to prove that a symmetrical diametral equation is of the same form when 

 transformed to secondary axes, it is only necessary to write the generalised form of the 

 expression in lines and columns. As the original axes are always supposed rectangular, 



