the Differential and Integral Calculus. 29 



" But if u be a function of two independent variables x and 

 y, then, because d u — d x udx + d y u dy> 



. . du j dy 



we have d x u = -3 — — d«u ~-. 



x dx y dx' 



from which it appears that the differential coefficient d x u can 



no longer be represented by the fraction -? — . Hence the 



only notation which is inconsistent with itself, and consequently 



erroneous in principle, is precisely that which is most used in 



this University, viz. the representing differential coefficients by 



- . du d* u 

 fractions-^ — , -=— ^ 



d x dx 1 



" This ought to be a sufficient reason for rejecting it and 

 endeavouring to invent another which shall at least be con- 

 sistent with itself. ********* 



" The ridiculous subterfuges to which writers have been 



1. ., r du d?u ,. „ 



driven by the use 01 -z — , -j- % ...render it a matter of won- 



der that that notation has not long ago been banished from 

 every mathematical treatise." 



The writer concludes his work by giving the following ex- 

 amples of the confusion arising from the common, which he 

 terms inconsistent notation. 



" At page 175 of the Cambridge translation of Lacroix's 

 Differential and Integral Calculus, we have these two equa- 

 tions : 



du du dz i- , du du dz 



dx dz dx dy dz ' dy 



on which the author remarks, * the d z of the first equation 

 must not be confounded with the dz of the second.' Now we 

 ask, what is there to distinguish dz in the one from dz in 

 the other ? Nothing. In fact, this remark alone ought to have 

 been sufficient to demonstrate the necessity of an improve- 

 ment in the notation. A little below in the same page, we 

 find the two following explanatory equations : 



7 d* j j 7 dz , 



dz — -j— . d x 3 and d z = -z— . d y, 

 d so it y 



which we hold to be utterly unintelligible, though they are 

 given by way of explaining the mystery of their predecessors 

 above. 



" We shall take our next example from a book, the title 

 of which it is not necessary to mention. 



11 ' z being a function of x and y, two independent quantities, 



