IN HYDRODYNAMICS, 33 



4. Since u = N — , v = N -^ , and w = N — ^ , we readily obtain from equation (2), 



dx ay ax 



^^Ar(^%^\^)=0 (3). 



dt ydx' dy" dx'l ^ ' 



which equation determines N. 



A remark is important here. It appears from the equality (l), that udx + vdy + wdz 

 = Nd\l/, and it might hence be supposed that when the left-hand side of this equality is integrable, 

 we are at liberty to assume JV = 1, and to consider \j/ identical with the quantity which is usually 

 called d) in Hydrodynamics. But it is clear from the reasoning by which equation (2) was 

 obtained, that iV is a quantity of the same kind as the velocity, and that \// is supposed to be 

 freed from any factors which do not verify the equation \^ = 0, whilst d(p is merely a substitu- 

 tion for udx + vdy + wdz, and its integral is subject to no such operation. It is not, 

 therefore, allowable in any case to suppose the two quantities to be the same, on which account 

 I have here employed the letter >// in the place of the (h of my former paper. When udx + vdy 

 + tvdz is integrable, in general N =f (t) . F ((f>). 



5. For the purposes of the reasoning on which we shall presently enter, it is required 

 to shew, first, that when udoc + vdy + wdz is an exact differential (dcp), the integral of the 

 dynamical equation may be taken from any one point of the fluid to any other, and that the 

 arbitrary quantity to be added is either a constant or a function of the time only. This will 

 appear as follows. 



The general dynamical equation is equivalent to the three equations, 



dP „ /du\ , ^ dP ^^ fdv\ ^ . dP „ {dw\ 



d?-^n^)=''^*^- rf^-^nTj=°'<^)- di-^nd7)=°'f^> 



in which P is substituted for — , or for k^ Nap. log p, according as the fluid is incompressible 

 or compressible. Assuming JTdx + Vdy + Zdz to be an exact differential, putting (dX) for 



I X\dx + I — V) dy + l- Z\ dm, and adding the above equations after multiplying 



them respectively by dx, dy, dss, it is known that we obtain for the case in question, 



<->^(-^t)^i(-|S'^f^^4)=°- «■ 



,.,.<. xro , , . , , d0' rf0' d0' . 

 which, 11 V- be substituted for — -=— + -^ + —— , is equivalent to 



dx' dy' dx' 



(d\ d'd, „dr\ (d\ d^(i> ,,dV\ id\ d'<p „dV\ , ^ 



\dx dxdt dx I \dy dydt dy j \dx dzdt dx I 



But the quantities in brackets must be respectively identical with the quantities on the left-hand 

 sides of the equations (4), (5), (6). Hence by reason of those equations, 



d\ ^d) dV d\ d^d) , dF d\ d'rf) dV 



— + — ^-- + V — = 0, — -(- — 2_ + F — = 0, — + — J- +V — = 0. 

 dx dxdt dx dy dydt dy dz dzdt dz 



Hence, dividing the foregoing equation by dx, it will be seen that — and — may be of any 



arbitrary values. The integral of that equation may consequently be taken from any one point 

 to any other of the fluid, and the arbitrary quantity to be added is independent of co-ordinates. 

 Vol. VII. Part I. E 



