OF FLUIDS ON THE MOTION OF PENDULUMS. [59] 



In this expression p denotes, in the case of an elastic fluid, the pressure statically corres- 

 ponding to the density which actually exists about the point whose co-ordinates are x, y, z, and 

 the part of the expression which contains p denotes twice the work converted into vis viva in 

 consequence of internal expansions, and arising from the forces on which the elasticity depends. 

 The last part of the expression is essentially negative, or at least cannot be positive, and can 

 only vanish in one very particular case. It denotes the vis viva consumed, or twice the work 

 lost in the system during the time dt, in consequence of internal friction. According to the 

 very important theory of Mr Joule, which is founded on a set of most striking and satisfactory 

 experiments, the work thus apparently lost is in fact converted into heat, at such a rate, that 

 the work expressed by the descent of 772 lbs through one foot, supplies the quantity of heat 

 required to raise 1 lb. of water through 1° of Fahrenheit's thermometer. 



50. The triple integral containing n can only vanish when the differential coefficients of 

 u, v, w satisfy the five following equations, 



du dv dw 



dx dy dz ' I 



dv dw dw du du dv 



— + =0, +=0 , — + — = 0. 

 dz dy dx dz dy dx 



(137) 



These equations give immediately the following expressions for the differentials of u, v, w, 

 in which the co-ordinates alone are supposed to vary, the time being constant : 



du = $dx — (a'"dy + w"dz, "j 



dv = $dy - w'dz + w"'dw, > (*38) 



dw = $dz — w"dx + tody. ' 



In these equations \ w, w", w" are certain functions of which the forms are defined by 

 the equations (138), but need not at present be considered. It follows from equations (138) 

 that the motion of each element of the fluid within the surface S is compounded of a motion of 

 translation, a motion of rotation, and a motion of dilatation alike in all directions. So far as 

 regards the first two kinds of motion, the fluid element moves like a solid, and of course there 

 is nothing to call internal friction into play. For the reasons stated in my former paper, I was 

 led to assume that a motion of dilatation alikein all directions, (which of course can only exist 

 in the case of an elastic fluid,) has no effect in causing the pressure to differ from the statical 

 pressure corresponding to the actual density, that is, in occasioning a violation of the func- 

 tional relation commonly supposed to exist between the pressure, density, and temperature. 

 The reader will observe that this is a totally different thing from assuming that a motion of 

 dilatation has no effect on the pressure at all. 



When the fluid is incompressible S = 0, and it may be proved without difficulty that 

 W, to", to" are constant, that is to say, constant so far as the co-ordinates are concerned. In 

 this case we get by integrating equations .(137) 



u = a — to"y + to'z, i 



v = b - to'z + w'"x, \ (139) 



w = c - to'x + to'y. ' 



32—2 



