160 DIFFERENTIATION OF 



Hence the differential coefficient of ( -7-) -~ is 



\dyj ax 



tfu\ dy\dj, fdu\ $y_ 



\\dxdyj \dy z j dx) dx \dyjdx*' 



Also the differential coefficient of ( -7- ] is 



\dx/ 



f d~u \ du fd*u\ 

 I y JL. i I 



\dy dx) dx \dafj 

 Collecting the terms in (1) and (2), we have 



,. , 



4. 9 4. 4- =0 



\dx*J "* (dx dy) dx "*" \df) \dx) * \dy) dx z 



184. It is not necessary to proceed further with the 

 successive differential coefficients of implicit functions, as the 

 equations become too complicated to be often used. The 

 reader may, as an exercise, obtain the following result from 

 equation (3) of Art. 181, by either of the methods we have 

 used in Arts. 182 and 183 : 



\ dy_ f 



_ 



s * ^ 



_ 

 \dx s \dafdy dx \dxdy 2 \dx "" \dtf \dx 



( f d*u \ (d?u\ dy\ d*y (du\ d*y _ 



T w X 1 "I T~ I ~T I 3~~ ; ~T~ r ~j~~x T I ~J~ I ~Y~s " 



[\dxdyj \dy J dx) day \dyj dx 



We may observe that it is often found convenient to use a 

 certain abbreviated notation for partial differential coefficients. 

 Thus \i$(x,y] be any function of a; and y, any partial differential 

 coefficient of the function may be indicated by the letter <f>, 

 with accents above corresponding to the number of differen- 

 tiations with respect to x, and with accents below correspond- 

 ing to the number of differentiations with respect to y. For 



example, <f>" will indicate ( . ' ) , and </ will indicate 



V dx J 



x, y) 



\ 



, 

 J 



and so on. 



, , 

 V dxdy 



We may also use y for ~ t and y" for -=-f , and so on. 



dx dx 



Thus with the present notation the equations (1) and (3) of 



