SOLUTIONS OF HOMOGENEOUS AND CENTEAL EQUATIONS. 1045 



plane curve or central surface, while others of these quantities may represent the 

 coordinates of the centre. If now the function represents a central surface whose centre 

 is variable in position, we may have seven quantities, whereof one is a parameter, and 

 six are variables. When the number of variables exceeds three there must, of course, be 

 other relations between the coordinates, otherwise the problem becomes indeterminate. 

 In the case supposed, a second equation between the coordinates of the centre and one 

 of the coordinates of the surface determines the curve or surface which the centre is 

 supposed to describe, and supplies the necessary elements for the solution of the first 

 equation. I have introduced this illustration because every homogeneous equation of 

 even degree of three or four quantities represents a central curve or central surface 

 respectively referred to the centre ; and it is easily seen that, if the origin be changed to 

 any point, whether exterior or interior to the curve, the left-hand side of the resulting 

 equation is a homogeneous function of the original coordinates, and the coordinates of 

 the centre. 



2. Application of the Principle to finding Solutions of Homogeneous Equations 



of one part. (Case I.) 



The most obvious application of the method of homogeneous variation is to the 

 exact determination of a series of points on a curve or surface whose equation is given 

 in the form of a homogeneous function equated to an arbitrary term. The method, 

 however, is purely analytical, and it is not necessary that the quantities should have a 

 geometrical interpretation. The arbitrary term is to be expressed as the n th power of 

 a number iv, and the equation is then of the form- — 



x n + A 1 x n ' 1 y + A 2 x n - 2 y 2 + . . . +A n y n = w n . 



The quantity to is evidently a parameter, being the value of x when y = 0. It is 

 required to find a series of exact values of x and y to the given parameter w. The 

 values to be found may be denoted by x u y u x 2 , y 2 . . . Let £ 1} ^ be any values arbitrarily 

 assumed; these values are to be inserted in the given function, and the value of the 

 parameter computed by summing the terms and extracting the nth. root of the sum. 

 The equation formed may be called auxiliary equation (1); and may be written — 



Then by the preceding theorem we have the relation xj^ = y\\y\\ = wjw u which gives 

 for the coordinates of the first point (or first set of values of the original equation) 

 x 1 = ^ 1 wjw 1 ; y 1 = t) 1 wfw 1 . 



A second auxiliary equation being formed from new assumed values ^ 2 -q 2 , and the 

 parameter w 2 computed, we find from these data the coordinates of a second point (or 

 second set of values of the original equation), viz., x 2 = £ 2 w/w 2 ; y2 = r) 2 iv/yj 2 , and so on. 

 These are true algebraic solutions of the given equation. 



