Trans/ui ina/ion of Fuiiclions. 1 7 1 



Ax +... + K.r_| j^eVv''-'+eQv^'-'+ +m. ' ^"^l 



Add L to each side of this equation, and the sum is the equa- 

 tion 



But since the first side of this equation (D) is the same as the 

 first side of equation (B), and the powers of v on the second 

 side of this the same as the powers of v on the second side of 

 equation B, the coeflicients and absolute number of the se- 

 cond side of this equation must be respectively equal to the 

 coefficients and absolute number of equation B : whence 

 P'=P, Q'=Q + cP, R' = R+eQ&c. and /3' = L + ^«. 



Corollary 1. Hence in any two consecutive equations, of 

 which the first is one degree lower than the second, any co- 

 efficient of the second side of the second equation is equal to 

 the corresponding coefficient of the first equation plus the pro- 

 duct of the next preceding coefficient of the first equation and 

 the quantity e. 



Corollarij 2. Hence if any number of equations are derived 

 from each other, the absolute number of the second side of 

 any equation is equal to the absolute number of the first side 

 of the same equation plus the product of the absolute number 

 of the second side of the next preceding equation, and the 

 quantity c. 



Corollarij 3. Hence if any number of equations are derived 

 from each other, the coefficient of the first term or highest 

 power of T' of the transformed equations is the same in all. 

 Hence in the following consecutive equations 

 A = P 

 Ax+B = Pt^-hB, 

 Aa;^ + B.r +C = Vv' + B.v + C, 

 A.r M B.z^ + Ca + D = Pi'' + B f^ + C,v + Da 

 &c. &c. 



By corollary 1 we derive 



B,= B, + rP 

 B.=B,4-a' 



&c. 

 C,= C,+^B, 

 &c. 

 By corollarv 2 we derive 



B.-B-h'P 



C,= C + rB, 



D,= D-f-rC, 



&c. 



\' 2 By 



