480 ON PHYSICAL LINES OF FORCE. 



The sum 8W+8E must be zero by the conservation of energy, and &(xyz) = Q, 

 since xyz is constant; so that 



(61). 



In order that this should be true independently of any relations between a, /3, 

 and y, we must have 



Sa = a^, S0 = j8^, Sy = y^ ...................... (62). 



x y z 



PROP. X. To find the variations of a, ft, y due to a rotation 6 l about the 

 axis of x from y to z, a rotation t about the axis of y from z to x, and a 

 rotation O t about the axis of z from x to y. 



The axis of ft will move away from the axis of x by an angle 0, ; so 

 that ft resolved in the direction of x changes from to ft0 t . 



The axis of y approaches that of x by an angle 6 3 ; so that the resolved 

 part of y in direction x changes from to yd*. 



The resolved part of a in the direction of x changes by a quantity depending 

 on the second power of the rotations, which may be neglected. The variations of 

 a, ft, y from this cause are therefore 



Ba = y0 t -ft0 t , 8ft = a0 i -y0 1 , Sy = ^-a^ ............... (63). 



The most general expressions for the distortion of an element produced by 

 the displacement of its different parts depend on the nine quantities 



d d d ~ d ~ d ~ d ., eZ~ d d -, 

 -j-ox, -3- ox, -j-Sx; -j-oy, -j-oy, -j-oy; -r-oz, -y-8z, -5-02; 

 dx dy dz dx y dy y dz y dx dy dz 



and these may always be expressed in terms of nine other quantities, namely, 

 three simple extensions or compressions, 



Sx' 8y' Sz; 

 x" y" z- 



along three axes properly chosen, x', y", z', the nine direction-cosines of these 

 axes with their six connecting equations, which are equivalent to three inde- 

 pendent quantities, and the three rotations # & S about the axes of x, y, z. 



Let the direction-cosines of x' with respect to x, y, z be l a ra,, n lt those of 

 //', I., in,, n a and those of z', l n m^ n t ; then we find 



