76 ON THE ALGEBRAICAL SOLUTION OF 



PART III 

 1. The equation 



has been shown by Euler l to be insoluble when n=2, and 

 there is reason to believe that it is also insoluble when 7i=3, 

 although no demonstration has ever been given. 2 The case 

 n=4: does not appear to have been solved either algebraic- 

 ally or otherwise, but the present writer has discovered one 

 numerical solution, which shows that the equation is soluble. 

 When n exceeds 4 there is no great difficulty in obtaining 

 algebraical solutions, but these are of a very specialised 

 character, on account of the particular assumptions made as 

 to the forms of the roots, and are not at all to be regarded as 

 typical of the general rule. 



It will be convenient to commence with the cases n=5, 6, 

 and 7 ( 2-4) from the formulae for which it will be shown 

 algebraical solutions for all values of n greater than 7 may be 

 deduced, then to proceed in the light of these results to discuss 

 the case 7i=4 (6), and finally to give algebraical solutions for 

 all values of n greater than 2 of the equation transformed by 

 replacing P 4 by P 2 ( 7). 



2. QUESTION 1. Solve the equation 



P<f=P 



We have identically 



and (x+y)*+(x-y)*+(2y)*=2(x*+3y*)*. 



1 Elements of Algebra, Fourth Edition, 1828, Part n., Chap. xiii. 206-208. 



2 Cf. Euler, Commentationes AritJimeticce, vol. i., xxxiii. 1 ; vol. n., Ixviii. 3. 



