Transformation of Functions. 169 



Demonstration. 



First let A = A 



(1) Multiply the first side by x and the second side by v + ?, 

 and . ( Av 



\ 4-Ae' 



Add B to each side, and 



Let B,=fAH-B, and 



A.r + B = Ar;+B, 



(2) Multiply the first side by x and the second side by X' + e, 

 and A o , T3 f Air + B|T' 



A^- + B.r=| +,Ar, + .B, 



Add C to each side, and 



x " . M , n i Av''+ B.t;+ C 

 A.r-fBa'+C= < , A ■ n 



( + eAv + t^B, 



Let B. = eA + B„ C,=t'B, + C, and 



A.r^ + B.r + C = At;' + B,t; + C , 



(3) Multiply the fii'st side by x, and the second by v + e, and 



' 14" ^Af' + eBy + eC, 



Add D to each side, and 



Ax+Bx- + Lx-{-D= < .- o, Tj- , r^ 



Let B, = ^A + B„ C,=^Bo + C, D,=^a+ D, and 



Ax ' + Bx- + Cx + D = Ad* -f Bju^ + C.v -f D3 : and so on . 

 So that having an equation of any degree, and its trans- 

 formed equation, which will be of the same degree, we shall al- 

 ways find the transformed equation of the next higher degree 

 by the verv same ste[)s from which tlie simple equation is de- 

 rived from the identical equation A = A. Therefore from the 

 identical equation A = A we transform the function Aa-f B 

 into Af + B,, or derive the simple equation At + B = Au4-B,; 

 I'rom this simple equation we transform the function A.r--f- 

 Bj + C into Air + oBjU+Ca, or tlerive the quadratic equation 

 A.r' + Bx + C = At>'4-B,y-|-C,, and so on, from one degree to 

 the next higher, till we arrive at the function required. 



Tlierefore, without proceeding further, the principle of de- 

 rivation will point out the general law. For this purpose col- 

 lecting the values of the substituted quantities, we obtain 

 B, = ^A-1-B 



B=/'A + B, C,=^B. + C, 

 B,=r^A + B, C,=eB,+C, D,=<?C,-hD, 

 &c. &c. 



where the law of derivation for the co-eflicients is obvious. 



Vol. C'2. No..'K\5. .SV;)/. 18'iy. Y Ex- 



