DIFFERENTIAL COEFFICIENT. '149 



, r . Aw . du 



the limit of -r is =,- , 

 A# dx' 



the limit of -^- is -- , 

 Ace cto 



,, T ., f &z. dz 

 the limit of -r is -y- . 



The limit of > -. ig ^ differential 



As 



coefficient of < (y, z) or w, with respect to z, taken on the 

 supposition that z is the only variable ; and may therefore be 



denoted by -y- . 

 J dz 



The limit of .-,* wouldj 



did not change, be the differential coefficient of <]>(y,z + As) 

 with respect to y. But as As diminishes without limit with 

 Ay, the limit is the differential coefficient of <j> (y, z), with 

 respect to y, taken on the supposition that y is the only 

 variable. 



We have then finally 



du _ du dy du dz 

 dx dy dx dz dx ' 



d\L 



170. In this result , denotes, as stated, " the differential 

 dy 



coefficient of u, taken with respect to y, supposing y alone to 

 vary" It is not impossible that the reader may be inclined 

 to say, " But y and z being both functions of or, if y varies, 

 z must vary too, how then can I make the supposition that 

 y alone varies?" His own further reflexion will probably 

 remove the difficulty, if such it be. Should he however be 

 unable to satisfy himself, it may be suggested to him that 

 we do not make the supposition that y alone varies as a 

 final supposition. We allow for the variation of both y and 

 z, but it is convenient for our purpose to consider these varia- 

 tions one at a time. 



T , . . , . (du\ , fdu\ . , f du ,du 



It is usual to write ,- and -y- , instead ot -,- and -j- , 

 \dy) \dzj dy dz' 



the brackets serving to remind us of the suppositions to .be 



