Motion of Solid and Fluid Bodies. 153 
here included among the external forces, as they are quite different from the 
molecular forces, and act at finite distances; and any statement made in reference 
to the function v is to be understood only of the molecular forces, which alone 
are included in it. Also, all such forces as those arising from the pressures of 
the sides of vessels containing a fluid, &c. are excluded from the function v. 
Let (p, a, B, 7) be the original length and direction of the line (mm’) joining 
two molecules ; suppose these to become (p+ p', a+ 0, B+ 6’, 7 +7), where 
(p’, a’, 6’, 7’) are the changes in length and direction ; it is evident that the force 
acting between m and m’ will be, 7m general, a function of (p, a; BY p's @’s Bs 7’): 
and also of (a, y, z); or, if (f) denote this force, it may be represented by 
S=F¥o +28 p +3r,p° + Ke. ; 
Aj + aa? + &e. 
+4 3,6 + 1,6? + &e. (2) 
Oy + cy” + &e. 
Fy Fy &C., Ay By C, &c., being functions of (p, a, 8, ys % Y, %) in general; and 
the part of (/) depending on (p’) arises from the change of distance ; while the 
part depending on (a’, f’, 7’) arises from the change of direction. If we con- 
sider, however, that the change of force produced by a change in direction is 
continuous, while an indefinitely small change of distance developes a finite 
molecular force, it will be evident that we may neglect the part of (/) depending 
on change of direction, together with the part arising from high powers of (p’); 
consequently 
F=*, +25,p + &c., (3) 
where F,, F,, &c., are functions of (p, a, 8, y), and implicitly of (7, y, 2), if exter- 
nal forces act on the system, or if the body be naturally heterogeneous. 
As the force (f) is a force tending to alter p4-p’, its moment will be f-<p' ; 
hence, neglecting the higher powers of p’, we shall have the sum of moments of 
internal forces at a given point, 
v= X( EE p" 4 Fo(p’)*) 5 
v = SSW" ew’ + Fp?) p'sinddpdods, (4) 
0 
and, finally, 
and 
v=Y+tV3 
v, arising from the terms F,, and vy, arising from the terms F,. 
y 2 
