ON DIFFERENTIALS. 393 



the limit of - j tuhen h is indefinitely diminished 



dij 

 is denoted by ~ . Some writers add the following words : the 



CL3C 



quantities dx and dy are called, the differentials of x and y 

 respectively ; their absolute values are indeterminate, and they 

 may be either finite or indefinitely small provided their relative 



magnitudes be such that -jr- is equal to the limit above men- 

 tioned. 



With this meaning attached to dy and dx such equations 

 may occur as 



dy = <f>' (x) dx, 



where </>' (x) is the differential coefficient of (x) or y. 



Equations expressed by means of differentials are in 

 general capable of immediate translation into the language 

 of differential coefficients. For example, if x and y be co- 

 ordinates of a point on a curve and be functions of a third 

 variable t, and if s denote the corresponding arc of the curve 

 beginning at some fixed point, we have, by Art. 307, 



dt 

 and by differentiation 



dx d?x dy d*y _ ds d"s 

 Tt df + 'dt df dtdf' 



A writer who uses differentials will express these results thus, 



dx d*x + dy d?y ds d's. 



The student may look upon the latter as merely abbreviated 

 methods of writing the previous equations, and may take 



/-7/-v /"/7/ /x"l/* 



dx, dy, d^x, ... as standing for -r , -jj- , -jy , . . . respectively. 



367. Let u be a function of any number of variables, 

 for example three, so that u = (j>(x, y, z). If we suppose 



