GRAPHICAL DIFFERENTIATION AND INTEGRATION. 1 29 



with f(x, y, z, t) . For this we can develop (c) according to Taylor's theorem, and 

 leave quantities of the second order out of consideration, (c) then takes the form 



/ (*, y, z, t) + Vx dt+v y dt+v,dt+fdt 



cx cy cz ci 



The excess df of the value of / at the point (x + v x dt, y + v y dt, z + vjdt) at the time 

 t+dt over its value in the point (x, y, z) at the time /, will then be 



df = 9 ldt+^v x dt+%v y dt + ^v z dt 

 J St Sx dy y ?z 



If we divide this equation by dt, we get a derivative which gives the rate of change 

 of the value of/ at one and the same moving material individuum. We shall call 



this the individual derivative, and denote it by . Its expression in terms of the 



dt 



four partial derivatives (b) will then be 



or in vector-notations 



A case of special importance is when / represents one component of a vector 

 A. The individual time-derivative of the vector A will then be expressed by the 

 three equations 



dA x ?A X . 9A X SA X . 9A X 



dA y _?A, 9A_ 9Ay SA, 

 ~Tt U +V 'l^ + ^3y +V 'dz 



dA. z cA z . c-aL 3 . c A z . c A z 



-it=ir +v ^ +v ^ +v ^ 



or, using the vector-notations introduced in section 1 74 



dk 3A , A 



w -di = -Ji +vVA 



An important case is that in which the vector A is the velocity of the moving 

 particle. The rate of change of its velocity gives its acceleration, for which we 

 thus get the equation 



dy cv . _ 



W ^T7 +vVv 



In order to form the field of acceleration we have thus to perform pure time-deriva- 

 tions and pure space-derivations, which we have investigated already. 



The distinction which we have here introduced between local and individual 

 time-derivations will be of great importance in our continued work. The difference 



