PEIRCE. 

 1 



DERIVATIVE OF A SCALAR POINT FUNCTION. 



351 



ii-v iti 



dufd 2 e dw d 2 v dw 

 dz\dz 2 dz dx-dz dx d 



dv dw\ 

 z~l)y'ljy) 

 2 du do/dk v dw dh v dw dh v dw\ . 

 K"dz' dz\dz ' ~dz + ~dx ' dx Jy ' ~dy ) * 



It is evident that D v D w ii is usually quite different from D w D v u. 



In the transformation of a partial differential equation from one set of 

 independent variables to another set which does not form an orthogonal 

 system, derivatives occur which are not normal in the sense of the last 

 paragraphs. If a mass of fluid is in motion under the action of given 

 forces, it is usually convenient either to express the orthogonal coordi- 

 nates of a particle which at the time t has the position (x, y, z) in terms 

 of t and the coordinates x Q , y , z , which the same particle had at the 

 origin of time, or to express x Q , y , z , as functions of x, y, z, t. 



*o =/i 0, V, *, t\ !/o =/a 0> V, z, t), z =/ s (x, y, z, t). (31) 



In this case, it frequently happens that the level surfaces of fi,f 2 ,fz, 

 are not orthogonal. According as we use the "historical" or the "sta- 

 tistical " method of studying the motion, we shall express the pressure 

 and the density in terms of x , y , z , t, or in terms of x, y, z, t. Sup- 

 pose the second method to have been chosen, and dp / dx to have 

 been found by the aid of Euler's Equations of Motion and the Equation 

 of Continuity, and suppose that dp / dx is needed. We shall then have 



dp _ dp dx dp dy dp dz 

 dx dx dx dy dx dz dx ' 



If with the help of (31) we find the values of the determinants 



(32) 



L = 



(33) 



and put 



Q= L -£ +N - d i +*■*£• 



B 2 = L 2 + M 2 + N\ 



