CHAPTER IX. 



LAGRANGE'S THEOREM AND LAPLACE'S THEOREM. 



138. SUPPOSE y = z + x<f>(y) ........................ (1), 



where z and x are independent, and it is required to expand 

 f(y) according to ascending powers of x. Put u for f(y], 

 then, by Maclaurin's theorem, we have 



" 



CM , , M 



where , -r, -3-3, ... denote the values of u, -, , -^- 2 , ... 



when x is put = after differentiation. We proceed to trans- 

 form these differential coefficients of u with respect to x into 

 a more convenient form in order to ascertain their values 

 when x = 0. We shall first shew that 



d (, ,dv\ d (, .dv\ . . 



4-P-w-rr = -j-\F( v ) j-r ............... ( 2 )' 



dx \ ^ J dz) dz\ ' dx] 



supposing that v is any function of the independent quantities 

 x and z, and F(v) any function of v. 



To establish (2) we need only observe that the left-hand 

 member is 



., , . dv dv r, , N d?v 



and the right-hand member is 



-n, , .dv dv -nf N d?v 

 F' (v} -T- ^-+ J^() j r 5 

 ^ ' dxdz ' dz dx 



and these two expressions are equal by Art. 134. 

 From (1) we have 



