436 
MR. 0. HEAVISIDE ON THE FORCES, STRESSES, AND 
div Sftg = Fq + 2ftVg ;. (49) 
where the first term on the right is the translational activity, and the rest is the sum 
of the distortional and rotational activities. To separate the latter introduce the 
strain velocity vectors (analogous to Pi, P* P 3 ) 
Pi = i ( v 2 i + v iD, P 3 = i ( v ?2 + v 2 l), P 3 = i(V ?3 +V 3 q); .... (50) 
and generally 
Pn = i ( V -9N + NV.q).(51) 
Using these we obtain 
2 avq = a lPl + a,p 3 + a 3 p 3 + ^ + a 3 YzrzM 
= 2 ftp + \ CljVi curl q + \ CLVj curl q + \ ft 3 Vk curl q 
= 2 ftp + € curl q .(52) 
Thus 2Qp is the distortional activity and € curl q the rotational activity. But 
since the distortion and the rotation are quite independent, we may put 2Pp for the 
distortional activity ; or else use the self-conjugate stress, and write it ^ 2 (P + Q) p. 
§ T2. In an ordinary “ elastic solid,” when isotropic, there is elastic resistance to 
compression and to distortion. We may also imaginably have elastic resistance to 
translation and to rotation; nor is there, so far as the mathematics is concerned, any 
reason for excluding dissipative resistance to translation, distortion, and rotation ; 
and kinetic energy may be associated with all three as well, instead of with the 
translation alone, as in the ordinary elastic solid. 
Considering only three elastic moduli, we have the old k and n of Thomson and 
Tait (resistance to compression and rigidity), and a new coefficient, say %, such that 
€ = n Y curl D,.(53) 
if D be the displacement and 2€ the torque, as before. 
The stress on the i plane (any plane) is 
= » (VDj + V X D) + i (h - f n) div D + V curl D.i 
= (n + w x ) VjD + (n — «j) VDj + (h — § n) i div D ;.(54) 
and its conjugate is 
d, = » (VD X + V X D) + i (k -1») div D - n x (V X D - VD X ) 
= (n — 7? x ) VjD + ( n + n x ) VDj + i (k — § n) div D ; .(55) 
from which 
Fj = div ft 3 = (« — + It — f n) div D + (n + n : ) V 3 D 1 ..... (56) 
is the i component of the translational force; the complete force F is therefore 
