1056 HON. LORD M'LAREN ON SYSTEMS OF 



the original heterogeneous equations xy, £77, have been made homogeneous by introducing 

 supplementary powers of P and II, 



As the equations are homogeneous and identical in form, they represent similar 

 curves ; and according to the fundamental theorem of this paper the one form may be 

 derived from the other by multiplying every term by a constant, that is by (P/IT) (i . 

 Hence x, y, and P are obtained from £ 77, and II by multiplying each by the factor P/IT. 

 This can only be true if P and IT are the parameters or the same multiple of parameters 

 of the respective curves. 



It may occur as a difficulty that in the case of heterogeneous curves, the quantity P 

 does not correspond to the value x of x when y is equated to zero. But P can easily be 

 shown to be proportional to x . For suppose y and 17 in the two curves of the example 

 equated to zero, the equations are then of the form 



a£ + Ba£ _1 P + Caf- 2 P 2 =F =F = P" 



H + B^n + c^- 2 n 2 TT=ff. 



Dividing by the highest powers of P and II, we have 



t) +b (tv +g W =f=f=1 



Hence by the known law of expansions, x jY = £ /U, or the quantities P and II of similar 

 curves have a constant ratio to the intercepts x £ . They are therefore virtual parameters. 



(3) The case of a homogeneous equation of the n th degree equated to a term Z" or P n , 

 with which the paper commences, is now seen to be merely an explicit form of the general 

 conical or parametral equation, f (x, y, z) = 0. If the explicit term be considered as 

 a third coordinate (z), the conical surface is referred to a plane of symmetry, xy, and 

 an axis of symmetry z. In the implicit function the projections of the similar parallel 

 sections in the plane xy are neither similar nor symmetrical ; and the similar sections 

 are only found by taking z into account. 



So with the implicit function considered as of three quantities in one plane. The 

 parameter, P, is evidently not the principal parameter of the curve, but is the value of 

 the intercept on the axis of x in the system of axes proposed. 



From this investigation we see that any plane curve whatever may be expressed as a 

 homogeneous function of rectangular coordinates and the intercept on one of the axes. 

 When so expressed, it is a similar and similarly situated curve with respect to any other 

 curve expressed as the same function with a different value of P. 



