io6 A NETV SOLUTION' 



denominator of the radical, I find that it is always divifible by 

 M, the quantity in the numerator : and that thus the formulae 

 may be exhibited in another fliape, having this advantage, that 

 it will introduce fmaller numbers in the arithmetical operations 

 requifite for computing a and I;. 



1 WRITE QxM=3(BN— 3CM)'+2A(BN — 3CM)(3N— AM) 

 -{-B(3N — AM)% and evolving by adlual multiplication 

 QJK M = 3B^ N' — I SBC. MN + 27O. M^ 

 4- 6AB. N^ — 2A-B. MN 



— 18AC. MN + 6A^C. M' 

 + 9B.N'— 6AB.MN+ BA\M^ 

 Now, the coefficient of N- is, 3B- + 6AB + 9B = 

 3B (B + 2A + 3) =z 3B X M : dividing therefore by M, we get, 

 Q^=3B xN^ — (i8BC + 2A'B+i8AC + 6AB)XN 

 + (270^ + 6A--C + BA^) X M. 



And if for N and M we fubftitute their values (A + 2B + 3C) 

 and (3 + aA + B), we fliall find, 



Qj= 12B' + 81C'- — 54ABC+i2A'C — 3A'B' 

 all the other terms deftroying one another, except thefe five. 



All the terms in this value of Q^ being divifible by 3, 1 

 change Q^and put 



Q= 4B'+27C^ — 18ABC + 4A3C — A'B^ 

 or, Qj= (4B' + 27C^) — 2 AC (9B — A^ + A' (2AC — B^) 

 And fince 3QM is now equal to the denominator of the i-adical 

 in the preceding formulce for a and b, we have the following 

 new formula;, 



+ BN_3CM. 



, _4. .,N-AM . 



What 



