1090 HON. LORD M'LAREN ON SYSTEMS OF 



tion curves of single symmetry of different degrees. Again, by giving different values 

 to z in any of the derived equations, a series of contour-lines may be traced representing 

 a surface which is symmetrical about one axis. 



(2) An equation also represents a curve of single symmetry when it is of the homo- 

 geneous form u n \u n _ p - 1, and (1) the function u n is of even, and u n _ p of uneven degree, 

 and also (2) the terms contain only even powers of one of the quantities, y. Thus the 



equation 



x 6 + FxY + Pay = x 5 - Qx 3 y 2 + 'Rxy i , 



represents a non-central symmetrical equation, from which contour-lines of a derived 

 surface maybe obtained by substituting x' + zfor x and y f 2 + z 2 for y in the equation, 

 and finding values of x' and y' from x and y for any required value of z. 



I ought, perhaps, to refer here to the case of curves composed of factors ; but the 

 subject has been already fully investigated ; and I could scarcely hope to add anything 

 material to what has been found by writers of higher authority in these matters. I only 

 make this observation, that when equations expressed in terms of factors are expanded 

 in terms connected by additive or subtractive signs, it is generally necessary to give 

 alternative signs to some of the terms, consistently with the original equation, otherwise 

 the complete ovals will not be obtained. I have given an illustration of such a curve of 

 the 6th degree, in which the expansion of the terms of factors does not lead to alternative 

 signs, and which forms an elegant symmetrical closed curve having thirty-two inflexions 

 in its orbit. Its equation is 



(x 2 -l)(x 2 -2)(x 2 -3) + (y 2 -l)(y 2 -2)(y 2 -3) = l. (PI. IV. fig. 5.) 



20. Parabolic Limiting Forms. 



Considered as a section of the homogeneous central surface, a parabola of the n ih degree 

 is evidently a section of such a surface parallel to any tangent plane of the asymptotic 

 cone. For such a section the inclination of the asymptotes (which is the same as that of 

 a parallel section through the centre) vanishes, which proves that the curve is parabolic. 

 It is not quite correct to describe a parabola (as is sometimes done) as being a curve 

 whose equation wants the highest power of one of the variables. Homogeneous equations 

 are always central curves, although they may not contain the highest powers of both 

 variables. An equation in x, y, represents a parabolic curve when one of the variables 

 does not occur in the highest homogeneous part ; in other words, when the highest part 

 consists of a single term, y n , or is reducible to a single term by transformation of axes. 

 Because, by transforming to polar coordinates, and dividing by sin" 6, we see that when 

 sin 6 is equated to zero r becomes infinite, and that there are no asymptotes, because u n 

 consists of a single term. When in the equation of a parabola of any degree, u n consists 

 of more than one term, then since u n must be derived by transformation of axes from a 

 single term, y n (where y becomes px+qy) the homogeneous part, u n ought to be a 



