172 



Mr. P. Nicholson o?i the 



By the third corollary P is the same in all, and in the first 

 it is equal to A ; therefore instead of P on the second side we 

 have substituted A. 



Hence by the following distribution the coefficients of a 

 transformed function of any degree in x—e or x-\-e may be 

 derived from the given coefficients A, B, C &c. of another 

 function in .r of the same degree, observing to substitute A for 

 its equal P. 



B, = B+eA 



B,= B, + rA C.= C+eB, 

 Bj=B-ft-A C,=C,+cB, B,=B + eC, 

 &c. &c. &c. 



The manner of proceeding with the operation of transform- 

 ing the quadratic function Aa'- + B.t-f-C into another in x — <' is 



&c. 



B 



C {e 



eA+B =B, 

 eA + B. = B, 



7B,+C = C, 



Where A, B^ are the coefficients of the first and second 

 terms, and C^ the absolute number of the transformed quadra- 

 tic function. 



Again, the manner of proceeding with the operation ot 

 transforming the cubic function Ax' + B.r* + Cr+ D into an-, 

 other in .r— e is 



B 



D 



cA+B =B, 



^A + B, = B, t'B, + C=C, 

 A eA+B, = B, eB,+ a=C, ^^+0 = 0^ 

 Where A, Bj, Ca, D, are the coefficients and absolute num- 

 ber of the transformed function, and so on. But by making 

 the proper substitutions in this last cubic example, we have 

 Ae+B = B, 

 2A^+B=B, Ae»+ Be+C = C,J 

 3Ae+B = B3 3A^' + 2Bc+C=C3| Ae' + Be'+Ce-\-B = 'D, 



by which the original function A-r' + Bct'^ + C^r + D is trans- 

 formed to 



A(x-ey + {3Ae + B){x-ey + {3Ac' + 2Be + C){x-e) + {Ac^ + 

 Be' + Ce + B) 



Problem. 

 To transform any rational function of x into another which 

 shall have x+e instead of x. 



Supposing the powers of x to decrease uniformly by unity 

 from left to right. A, B, C &c. to represent the coefficients 

 and absolute number of the proposed function in the order of 

 the powers of x, as in the demonstration. 



Place 



