312 EDWARD SANG ON THE APPROXIMATION TO THE ROOTS OF 



The successive terms of the elementary progression are easily found to be 

 [0]-l 



W-p 



[2] = P 2 +2 



[3] = p 3 + 2pq +r 



[4] = p 4 + 3p 2 q +2pr + q 2 



[5] = p 5 + Ap 5 q + 3p 2 r + 3pq 2 + 2qr 



[6] = p 6 + 5p i q +4p 3 r + 6p 2 q 2 +6pqr +q 3 -\-r 2 



[7]=p 7 +6p 5 2 +5p i r + 10p 3 q 2 + 12p 2 qr +4:pq 3 + 3pr 2 + 3q 2 r 



[8] = p 8 +7p 6 q +6p 5 r +15pY +20p 3 qr +10p 2 q s +6p 2 r 2 +12pq 2 r +q i +3qr 2 



and the general form of the nth term of the progression having the initials 

 A , B , C , is, C being regarded as the zero term, 



[n-l]rA+{[n-l]q + [n-2]r}B + [n]C 

 or 



[n-2]rB + [n-l]{rA+qB} + [n]C . 



But the elementary progression alone suffices to determine the value of the 

 ultimate ratio 1 : x . 



This process is applicable directly only to equations having suitable coeffi- 

 cients. In the case of pure equations, those whose qusesitum is the cube root 

 of some number, the coefficients p and q are both zeroes, and the progression 

 becomes 



, , 1 , , , r , , , r 2 , , , r 3 , &c, 



which contains the truism that the ratio 1 : r ; r:r*; is triplicate of that of 

 which we are in search. 



In order so to change the form of an equation as to fit it for the application 

 of this method, we modify Lagrange's process in a manner which may be best 

 explained by examples. 



Let it be proposed to extract the cube root of the number 2. 

 In the equation 



x 3_0a;2_0a;-2 = 0, 



a . 



we may write -y in place of x, so as to give to it the form 



a z -0a 2 b-Qab 2 -2b 3 = O, 



in which, if b represent the side of a cube, a stands for the side of the double 

 cube. 



