and the Laws of Plane Waves propagated through them. 99 



The forces being thus divided into two classes, the formula will consist of 

 two parts, 



2 (Pep + Flp' + &c. ) + S ( Qlfj + Qlq' + &c.) = 0. 



I\ F, &c., denoting the external forces, and Q, (2',&c., denoting the molecular 

 forces. If {x, y, z) denote the co-ordinates of a point at any instant of the mo- 

 tion, and a virtual displacement {ix, Zy, iz) be conceived, these quantities will 

 be functions of four independent variables, which will be the initial values of 

 (x, y, z), and the time. Let (u', v', w') denote the accelerating forces ; hence 



^(P¥ + P'¥ + ^o.)=m(X-u')cx + {Y-v')iy + (Z-w')izldm. 

 I shall assume that the virtual moments of the molecular forces depend upon 

 the differential coefficients of (Ix, 8y, cz), by means of the following Unear 

 equation,* 



dx dy dz 



-I- 7? ^'^" J. P ^^^ _L /? ^^- 



dx dy ' dz 



Inserting these values into the equation of virtual velocities, and integrating by 

 parts, we find 



= W\p {X -u')ix + p{ Y- v') c^ + p(Z- w') cz\ dxdydz 

 m[fdP,dFdP,\/dQ,dQ,dQA(dR, dR, dPA -1, 



+ \\{ P{tx + CiCcy + E{oz) dydz, (1) 



+ \\{P^lx + Q,.ly + R.fiz) dxdz, 

 + l\(P,lx + Q,hy + R,Zz) dxdt,. 



The equations of motion being determined by the triple integrals, and the con- 

 ditions at the limits by the double integrals. The equations of motion formed 

 from this equation will be 



* The ground of this assumption is the fact, that molecular forces depend upon the relative dis- 

 placements of the particles, and not on their absolute displacements. 



o2 



