48 ON THE ALGEBRAICAL SOLUTION OF 



by (6) and (7), when substituted in (4), furnish the solution. 

 Moreover, since the original equation (f>(Xi, X z , . . ., X n )=Q 

 is homogeneous, we can make the solution integral 

 by multiplying the value of each of the roots given by 

 (4) by the algebraical quantity A' 3 and by the numerical 

 quantity which is introduced from the fact that the value of 

 x n is in general fractional ; and since A' 2 , A' 3 are integral 

 homogeneous functions of x lt x 2 , . . ., x n _ v it follows that 

 the solution presents the roots X lt X 2 , . . ., X n as rational 

 integral homogeneous functions of the third degree in n 1 

 variables x lt x z , . . ., x n _ r 



If any of the quantities a it a 2 , . . ., a n instead of being 

 numerical are arbitrary literal quantities, they will appear in 

 the final values for X lt X& . . ., X n as variables, and will 

 therefore alter the number of variables in, and the degree of, 

 the final solution. 



2. If the equation <=0 is not homogeneous, and integral 

 solutions be required, some care in the choice of particular 

 solutions and in a suitable preparation of < must be exercised 

 to secure this end. An example of this is given in Question 6 

 below. 



3. This process is naturally open to failure when the 

 equation under consideration admits only of solutions of a 

 certain type. An example of this is the equation x 3 +y 3 =2z 3 , 

 which admits only of solutions of the type (k, k, k), or of the 

 type (k,k, o), 1 and the application of the method furnishes 

 only the same type of solution. 



4. It is to be remarked that if <=0 is a homogeneous 

 cubic in three variables, the solution does not present the roots 

 as functions of two variables in accordance with 1 ; for 

 <=0 may be regarded as a non-homogeneous cubic in two 

 independent variables, and the solution will not present the 

 roots as functions of even one unknown, but is again par- 

 ticular, and being in general fractional and distinct from the 



1 Euler, Elements of Algebra, part ii. chap. xv. 247. Fourth edition. 1828. 



