SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1057 



6. Soluble Cases of the Homogeneous Function f (x, y, zY = Q. 



This function, as has been observed, represents a conical surface, being the asymptotic 

 cone of all the concentric and similar surfaces that can be formed by equating the same 

 function of x, y, z, to different arbitrary terms. 



Unless the equation contains a large number of terms, it can in general be solved by 

 taking arbitrary values of those two quantities which are most involved, and solving for 

 the one which is least involved. 



These solutions represent points on the conical surface, and if it is desired to obtain 

 such solutions in series, so as to represent a plane section or curve, they may be reduced 

 by division to the argument x=l, y=l, or 2=1 as desired. It is only necessary to 

 tabulate one such series ; because the surface is conical, and values of y and z may be 

 obtained to any other argument x = a, by merely multiplying the tabular values of y and 

 2 by a. Consider, for example, the equation of a homogeneous surface of this form, 



x s + Axhf + Bx s z 3 + Cx 2 z° + Cy 2 z 6 + By 5 z 3 + Axhf + y s = 0. 



Here the equation is symmetrical in x and y, but contains no powers of z except the 6th 

 and the 3rd. Accordingly, we may form an equation by assuming values x t and y lt and 

 then solving the quadratic equation in z 3 , the root of which, being extracted, is a solution. 

 That is to say, the values x lt y x , and z x thus found satisfy the equation. 



7. Solution of Homogeneous Equations of Functions of the Variables. (Case IV.) 



Assume 



Vi = ax p +by q + c ; v 2 = dx p + ey 9 +f; v 3 = .... 



Any homogeneous expression in v x v 2 . . . equated to an arbitrary term, or to another 

 homogeneous expression in v x v 2 ... of a different degree, can be solved by the methods 

 previously given. Values of v Y v 2 . . . being first found, we have then two simple equa- 

 tions for determining x p and y q in terms of these values, whence x and y are found. 



The original equation is of course heterogeneous when the quantities ax p + by q + c, &c, 

 are substituted in place of v, &c; and by means of this new application of the funda- 

 mental theorem, an endless variety of heterogeneous equations may be formed and 

 solutions in series obtained. It is evidently a condition of the possibility of solving such 

 equations that the number of factors v x v 3i &c, shall not exceed the number of constituent 

 quantities, x, y, &c, of which values are to be found. 



If the indices p, q, &c, are even, the curve or surface is central; but the converse 

 does not necessarily hold. Thus v x = ax + b ; v 2 = cy + d, gives a central curve from an 

 excentric origin, of which PL VI. figs. 5 and 6 are illustrations. If one of the quantities, 

 t'i, be taken — a+ Jx the curve will only have single symmetry. 



If we take v\ = x 2 + z 2 ; vl = y 2 + z 2 , thus a series of values of x and y may be found 

 to an invariable value of z, and the series of points so determined will trace out one or 



