456 Prof. Young's Method of determining the Function X 2 . 



before adverted to, will show the practical advantage of these 

 precepts. 



1. Let the proposed equation be 



x 3 —5x 2 + 8x— I =0; 

 then the functions X, X! are, 



X = x 3 - 5.t 2 + 8* - 1 

 X! = 3x 2 — 10* + 8, 



and consequently X 2 = 2* — 31. 



Here, by help of the foregoing rule, the function X 2 is 

 obtained immediately from the two preceding, independently 

 of any bye operations : thus 2 * is found by multiplying 8 x 

 by twice 3, or 6, and subtracting — 5 times — 10* from the 

 result, changing the sign of the remainder; —31 is obtained 

 by multiplying — 1 by three times 3, or 9, and subtracting 

 —5 times 8 from the result, changing the sign of the re- 

 mainder, as before. 



2. Let the equation be 



x* — 2 x 3 — 7x 2 + 10* + 10 = 0; 

 then we have 



X = x 4 - 2 X s — 7x 2 + 10* + 10 



X x = 2 ** - 3 * 2 — 7 * + 5 



and consequently 



X 2 = 17^—23^—45. 



In this example the coefficients —2 and 2 are rendered 

 equal, and of the same sign as the latter, by means of the 

 factors — 1 and 1 ; employing these, therefore, instead of the 

 coefficients themselves as multipliers, we obtain 17* 2 by mul- 

 tiplying — 7* 2 by twice 1 or 2, and subtracting —1 times 

 3 * 2 , changing the sign of the remainder ; the next term is 

 derived from multiplying 10 x by three times 1, or 3, sub- 

 tracting — 1 times —7x from the result, and changing the 

 sign; and the last term is got by multiplying 10 by four times 

 1, or 4, subtracting —1 times 5, and changing sign. 



3. Lastly, let the equation 



2X 4 — 13* 2 + 10* — 19 = 



be proposed, in which the second term is zero. Then we 

 have 



X = 2** + 0* 3 — 13** + 10* — 19 



Xj = 4^ + Ox 2 — 13*+ 5 



.•. by precept (B.) 



X a = 26 x 2 - 30* + 76 



or 

 13* 8 - 15* + 38. 



