Transformation of Functions. 1 73 



Place the coefficients A, B, C &c. in a line, and the pails of 

 the operation to be performed in as many other parallel lines 

 to the line of coefficients as the exponent of the highest power 

 of .r has unity, calling them the first, second, third, &c. lines, 

 as they succeed the line of coefficients. Bring down the first 

 coefficient A immediately below A into the last line, which 

 will be the first coefficient of the transformed function. 



Multiply the coefficient A of the highest power into e; add 

 the product cA. to the second coefficient B, and })lace the 

 sum B, in the first line under the second coefficient B: add 

 the product eK to B,, and place the same B, under Bj pro- 

 ceed in this manner until all the lines have been used. 



Multiply B, into e; add the product <?B, to the next co- 

 efficient C above, and write the sum C, in the second line in 

 a column under C. Multiply B.^ into c; add the product 

 ^Bi to Co and place the sum C3 under C,. Proceed to find 

 the remaining members of the column so that one may oc- 

 cupy each succeeding line. 



Proceed in the same manner with each succeeding column 

 to the last line, which will contain the coefficients, and the ab- 

 solute number of the transformed function in the same num- 

 ber of members, and in the same order as the coefficients and 

 absolute number of the original function. 



Numerical Examples. 



Ex. 1. Transform the function 4j;' + 15.i'' — 6.r+15 into 

 another which shall have x — 3 instead of .r. 

 Operation. 



4> 



+ 15 



■6 



15 



(3 



124-15 = 27 



12 + 27 = 39 81— 6= 75 



12+39 = 51 1174-75 = 192 225+15 = 24-0 



So that the function 4.r'+l 5x^—6^+15 is transformed to 

 4(x-3)'-}-51(a:-3)'-+192(a— S)-h240; 

 or if X — 3 = u 

 4.r'+15.r^-6j.-+15 = 4-i;'-f-51t;'+192T;+240. 

 But as the minor parts of the process of multiplying and 

 adding arc so easily performed mentally, the results may 

 only be set down, which will save a great deal of writing, and 

 render long operations much clearer; and thus the work now 

 executed will stand thus, 



4 -\-\5— 6-f- 1 5(3 



. +27 



+39+ 75 

 4 +51+1924 210. This 



