APPLICABLE TO THE MOTION OF FLUIDS. 389 



the required equation is found to be, 



tdtf d<p 2 d(p s \ ld<]> d 2 <f> d<j> d 2 (j> d<p d 2 (p d<j> cPft d±d*<p d(j> d'(p \ 

 \dx i dy* dsfjydt'dx* dx'dxdt dt'dy 2 dy'dydt dt' dz' dz'dzdtj 



d<f> (dp d*(j> d<j? cNp dcjf d 2 ^ 

 ~ Hi' \da? ' da? dtf ' dy 2 + Hz 1 ' dz* 



ety d<}> d*<p g d$ dtp d^ ^dQ d$ d 2 <p \ = Q 

 dx' dy' dxdy ' dx ' dz ' dxdz dy' dz' dydzj 



According to the views maintained in this Essay, the above is the 

 equation that should be employed in the Theory of the Tides : but it 

 is probably too complicated to be available for that purpose. However, 

 the simple equation, 



is integrable at once, and gives V = ^— . And as, from what is shewn 



in Art. 14, the variation of V at a given point is the same as if r and 

 r were constant, 



dV <p\t) „ .„ , , . c dV , d>'(t) AT . /, h 



-dJ = W' Hence if r.= r + A, fe dr. - - 5^i Nap. log. (l + - 



Consequently by substituting in equation (14) 



F^t) + P-Q - ^Nap.log. (l+|) -+ £= 0. 



It would be beside my purpose to inquire now into the applications 

 that may be made of this equation. 



16. A differential equation in which the principal variable <j> is a 

 function of x, y, z, and t, might also be obtained for the case of a com- 

 pressible fluid, but it is of so complicated a nature that no use could 

 be made of it, and I therefore omit writing it down. It is important 

 to observe, that for curvilinear motions this equation rises to the third 

 order. The inference to be drawn from this circumstance is, that the 

 forms of the surfaces of displacement are entirely arbitrary in the general 

 case of curvilinear motion, the three arbitrary functions which the com- 

 plete integral contains, having to be determined by given conditions 



Vol. VII. Part III. Xx 



