SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1055 



duced which will make the equation a homogeneous function of P and the other two 

 quantities, x and y. Then dividing by P n , x and y are found for z = a ; P = 1. In this 

 way a series of values of x and y may be found to any argument z. The choice of the 

 method will of course be determined by the possibility of solving the equation in P. 

 Unless the degree of the equation is very high, or the terms very numerous, it will 

 generally be found that an equation can be formed from which P may be determined, 

 and the corresponding values of x, y, and z deduced by homogeneous variation. 



As in the case of plane curves, we see that in the case of surfaces also, any surface 

 may be expressed as a homogeneous equation of the three variables and the intercept P 

 on one of the axes. Also, any function of identical form, with a different value of P, is 

 a similar surface. 



The method is evidently capable of extension to equations of any number of unknown 

 quantities. 



There are two distinct geometrical interpretations of the processes here given, 

 according as we consider the new quantity z as being in a different plane from x, y, or iu 

 the same plane. 



(1) In the former case z is a third coordinate, and the 3-dimensional homogeneous 

 equation y (a?, y,z) = always and necessarily represents a conical surface. This maybe 

 proved (without drawing on the methods of the differential calculus) by transforming the 

 equation to cylindrical coordinates. XY is the reference plane in which r and 6 are 

 measured ; z is then perpendicular to that plane. Then writing r cos for x, r sin 6 for 

 y, we have a homogeneous equation in r and z with trigonometrical coefficients. Accord- 

 ingly if r and z be varied, while 6 remains unchanged, we have by the introductory 

 proposition r^z x = r 2 /z 2 = r 3 /z 3 , &c. This can only be true if r and z are coordinates of the 

 same generating line, which of course lies in a plane passing through the axis of z and 

 making the angle 6 with the plane XZ. More simply, as the result of the transformation 

 to cylindrical coordinates is to form a homogeneous function, f(r. z) = 0, this is known to 

 be the equation of two right lines, and the surface is then shown to be made up of 

 generating lines passing through the origin, which is the definition of a conical 

 surface. 



In order that the equation f(x, y,z) = may have real solutions, the highest power of 

 one of the quantities must be negative ; and it is easy to see that the homogeneous func- 

 tion of the n th degree,f(x,y,z) = is the asymptotic cone of all the concentric and similar 

 surfaces which can be found by equating the same function to an arbitrary term P". It 

 is in fact the limiting form of this series of concentric and similar surfaces when the 

 parameter P vanishes. 



(2) I began by observing that we might conceive the quantity z (which was intro- 

 duced for the purpose of rendering the equation to be solved homogeneous) as being 

 either in a different plane from x and y, or in the same plane. If it is considered as 

 being in the same plane, it is the parameter of the non-homogeneous curve, and may be 

 denoted by P. The proof is as follows : — Compare the two subjoined equations, in which 



