372 PROFESSOR CHALLIS," ON THE DIFFERENTIAL EQUATIONS 



in brackets they are complete, the variation being with respect both to 

 the time and the three co-ordinates. A differential, as {dp), is put 

 in brackets to indicate that the variation is with respect to the three 

 co-ordinates, the time being given. 



By substituting P for a? Nap. log p, and (dQ) for Xdx + Ydy + Zdx, 

 regarding X, Y, Z, as functions both of t and the co-ordinates, the 

 equation (2) will be changed to the following: 



w-w» + (£)*♦&)* + (&*-. «o. . 



2. That the above equations may be available for application to 

 proposed instances of motion, it is required to derive from them a 

 single partial differential equation in which the principal variable is 

 a function of x, y, z, and t. This has been long done on the par- 

 ticular hypothesis that udx + vdy + wdz is integrable per se. I pro- 

 pose to give some consideration to this case, preparatory to the more 

 general investigation that will follow. 



On the above hypothesis we may assume <p to be a function of 

 x, y, z and t, such that, 



(d(j>) = udx + vdy + wdz. 



Consequently, 



_ d(p _ d(p dip 



dx' dy' ■ dz ' 



3. If the fluid be incompressible p is constant, -j- = 0, and equa- 

 tion (1) becomes, 



du dv dw _ 

 dx dy dz 



Hence for this case the required partial differential equation is evidently, 



d l± + *± + $£ = o. 



dx" dy 2 dz 2 



I proceed to make a transformation of this equation which will 

 be serviceable in the subsequent calculations. 



