OF AN IMPLICIT FUNCTION. 157 



Hence, from (1), 



v\ dy AMI _ {(d\ <fy 

 dyj dx \dx)} ]\dy ) dx 



w 

 dv\ fd'u 



Now 



the latter symbol denoting that u is to be differentiated twice 

 with respect to x, on the supposition that x alone varies; also 



'do\ _ f d*u \ 

 dy) \dy dx) ' 



the latter symbol denoting that u is to be differentiated with 

 respect to x, supposing x alone to vary, and the result with 

 respect to y, supposing y alone to vary. Similarly 



faw^ 



dx \dxdy' 

 dw 



Hence, substituting in (2), we have 



/du\ (/ d*u \ dy f^u\[ _ (du\ (f<Fu\ dy 

 d*y \djj) \\dylte) fa + \&?)} ~ (fa) \\dif) 'dx 

 dx* ~~ fdu^ 



\dy) 



n ...................... (3). 



GfU 



If we substitute in (3) the value of -r given by (1), we 



c u \ ( u \ 

 d^) = (dxTyJ 



f<Pu\ (du\*_ 9 ( d*u \ fdu\ fdu\ fd?u\ fdu 

 ^y^ \dx*J (dy) . (dxdy) (dx) (dy) + \djf) \dx 

 x 3 (du\ 3 



W 



