ON DIFFERENTIALS. 395 



Aar, Ay, and Az do so. In this manner we may deduce from 

 (3) the equation 



where a, /3, 7, all diminish without limit when Aor, Ay, As 

 do so. If then du, dx, dy, and dz, denote quantities whose 

 absolute magnitudes are undetermined, but whose relative 

 magnitudes are those to which Aw, A, Ay, and Az, respec- 

 tively approach as their limits when they are all indefinitely 

 diminished, we have 



-. r - 

 dxj \dyj ' \dz 



Having thus established (2), we give an example of its 

 application. Since in establishing (2) we had no occasion to 

 consider whether x, y, and z, were independent or not, the 

 result is universally true, whatever relation be given or sup- 

 posed between the variables. If, for example, <f> (x, y, z] is 

 always = 0, we have 



fdd>\ j /d(b\ , /eN 



1-r }dy + \^r \ds = Q ............ (5). 



\dyj \dzj 



- 

 dx 



Now if ^> (x, y, z) = is the only equation connecting x, y, 

 and z, we may if we please vary x and z without changing y. 

 Hence in the preceding investigation Ay = throughout, and 

 therefore in (5) dy = Q; thus we have 



fty. 1 "" ' \ J~ I '*"' ~ V W' 



<?2 



Hence 





d^\ 

 (dz) 



where -7- is the differential coefficient of z t supposing x to 

 dx 



vary and y to be constant. See Art. 188. 



368. It would occupy too much space if we were to pro- 

 ceed further with the subject of differentials. Differential 



