SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1053 



n, Ti—1, n — 2, &c. The second form is a homogeneous surface equation, from which 

 (1) may be derived by giving to z the value unity. Suppose £ and rj to be arbitrarily 

 assumed quantities, and that we can by any known method find a value of the third 

 coordinate £, which will make the equation zero, then, dividing by £", we eliminate £. 

 Thus, by division, the second of the above equations becomes 



The quantities outside the brackets are unity, and the quantities inside the brackets con- 

 stitute a solution or value of equation (1), where £/£ is a value of x, and iqjl, is a value of 

 y, while z as a separate quantity has disappeared. £ may be considered either as a third 

 dimension or as a variable parameter. 



Accordingly if £ can be found and the arbitrary equation formed, the solution is at 

 once obtained by dividing £ and 77 respectively by £. Let S n represent the numerical 

 value of the homogeneous part within the first bracket formed by assuming arbitrary 

 quantities £ 17 ; S„_ x is the numerical value of the terms within the second bracket (which 

 are all of the degree n—1), and so on, and the equation is 



S B +S B _ 1 f+S„_ 2 ^+S„_ 8 f+S„_4^+&c.=0 .... (3). 



It is easily seen that the possibility of solution does not at all depend on the degree of 

 the given equation, but upon the relative degrees of the terms of £, which it is necessary 

 to introduce. If the equation consists of only two homogeneous parts, suppose of the 

 9th and 3rd degrees, we have a simple equation to determinate £, as in this example 

 x 9 +A 1 x s y + . . . =je s +B 1 as 2 2/ + . . . &c, which may be written u 9 = u 3 . By introducing the 

 quantity z—\ this becomes 



{x 9 + A 1 x 8 y + ...\ = {x s + B 1 x 2 y + ...}z 5 



. {f + A^ + fec.} 1 / 6 

 S {£ s + B l f*i7 + &c.} 1 ' 6 ' 



Then by the introductory theorem we find 



This is the case already considered in section 3. Similarly, if the auxiliary equation in 

 £, 7), and £ contains only the first and second powers of £, we have a quadratic equation 

 between £ and the sums of the numerical terms of the assumed quantities, whence £ 

 may be found, and thence exact solutions of x and y. If the auxiliary equation contains 

 £ 2 and £ 4 or £ 3 and £ 6 , we have a quadratic equation to determine £ 2 or £ 3 , whose root may 

 then be extracted. Or, finally, there may be a soluble cubic or biquadratic equation in 

 £ or some power of £. 



If the equation contains an arbitrary term, this is equivalent to an additional 



