39 4t ON DIFFERENTIALS. 



x, y, z, all functions of a variable t, and for shortness put 



du -. dx _ dy ^ dz ~ 

 - = DM, -T- = Dx, -4- = Dy, = Zte, 

 c& eft <ft *' dt 



we have 



j -r 



dx) \d) dz 



/i\ 



(1). 



In works on the Differential Calculus, which use differentials, 

 we find an equation similar to the above occurring at an 

 early period, namely, 



du = dx+ dy+ dz .......... (2). 



' 



The introduction and use of this equation form the principal 

 difference between such works and one which, like the pre- 

 sent, uses only differential coefficients. To establish (2) the 

 following method is adopted. 



Let u= (}> (x, y, z), 



and u + AM = <f> (x + A#, y + Ay, z + Az), 



therefore 



AM = < (x + Arc, y + Ay, z + Az) < (x, y, z} 



^<j>(x + &x,y + &y,z + As) - <ft (x, y + Ay, z + Aa) 

 AOJ 



<ft (a;, y + Ay, ^+ As) - <fr (x, y, z + As) , 



Ay 

 [ <^> (a;, y, z + &z}-<j> (x, y, z} ^ ^ ^^ 



If Aa:, Ay, and Az diminish without limit, the quantity 

 <f> (x + Aa;, y + Ay, z + Az) <f>(x,y+ Ay, 



approaches the limit ( -^ ) . If then we put for this quantity 

 V&e/ 



+ a, we know that a diminishes without limit when 





