46 Prof. Young on some Forms of Quadratic Moduli. 



deduced ; but it is much simpler to obtain them directly, as 

 in the above instance. 



If in the general formula (3.) we put«= — 1, and b=l, 

 the first of your modular equations will result, viz. 

 (# 2 + y 2 + z 2 — xy — xz —yz) 

 x (a/ 2 + y n + z'^—x'y'-x'z'-y'z') 

 = X 2 + Y 2 +Z 2 -XY-XZ-YZ. 



Now with these values for a and b, our quadratic v 2 — av + b = 

 becomes 



2 



Substituting these values of v, or either of them, for « or for 

 /3, we have, for our linear factors, the form 



or, which is the same thing, 



and the imaginary coefficients may be taken as interpretations 

 ofyourijand £. Although only a quadratic has been em- 

 ployed to determine these, yet they are evidently the imagi- 

 nary cube roots of plus unity. You have taken those of minus 

 unity ; so that, in multiplying two such forms together, the 

 functions of x, y, z in our results would differ, as to signs, 

 in some of the combinations. If, however, the signs ofy and 

 z in the factors here proposed be changed, which change is 

 equivalent to changing the signs of their coefficients, these 

 coefficients will then be converted into the imaginary cube 

 roots of minus unity, and we shall both agree. 



The second of your modular equations really requires a 

 cubic. The general cubic formula Lagrange has investigated, 

 and given in the Additions referred to. Each factor of this 

 formula is of the form 



x 3 + ax 2 y + (a 2 — 2b)x' 2 z + bxy* + {ab — 3c)xyz 



+ (b 2 — 2ac)xz 2 + cy 3 -f acy*z + bcyz 2 -f c*z 3 , 



the cubic which leads to it being 



s 3 — as't + bs — c=0; 



and from this form, if we suppose a and b to be zero, and c 

 to be unit, your second equation results, viz. 



(x B + y 3 + z 3 — 3xyz)(a/ 3 + i/ 3 + z 13 - 3x'y'z') 



= X 3 + Y 3 +Z 3 -3XYZ; 



