= 0. 



= 0. 



12 Proceedings of the Royal Irish Academy. 



Eliminating hv, 8w, Sw^, Sw^, Sw, from the last six of these equa- 

 tions we have 



m . - mz . mx {Y + T'iJ.)St 



m my — mx . {Z + T'v) Si 



■ mz my a —h —g \L -\- T' [yv — z^t)\ '^t 



. -mx -h b -f {M+ T'{z\-xv)]dt 



mx . -g -f c \N-^ T'(xn.-yK)'\Zt 



fi V yv-zp., z\-xv, Xfi-y\ SV 



From the similar equations for the body £ we get the equation 



m' . . . m'z - m'y' {X' - T'\) St 



m' . —m'z' . m'x {Y' —T'tx)St, 



m' m'y' — m'x' . [Z' — T'v) St, 



. —m'z' m'y' a' - h' — g' {Z' — T'^y'v — z'/u.)} St 



m'z . -m'x -h' V -f {M' - T'{z'K~x'v)\St 



-m'y m'x . - g' - f c' {N' - T'{x'n-y'\)}St 



\ fi V y'v — z'/u. z'\ — x'v x'fji. — y'\ SV 



From these two equations we can obtain T' and S V. Substituting 

 the values so found in equations (a) we can get at once T and 

 8v, 8w, 8<j)„ Swy, Sui., and in a similar manner from the corresponding 

 equations for the body B, we can get 8u', Sv', Sw', SoiJ, Sw/, Sw.'. 



EouGH Constraints.' 



To find the initial motion when definite points in the body are 

 constrained to move on rough surfaces is, in general, rather difficult, 

 owing amongst other things to the fact that the force of friction at 

 each of the constrained points is in a direction opposite to that in 

 which it begins to move. We will investigate the comparatively 

 simple case of a single point in the body being constrained initially to 

 move on a rough surface. 



Take as axes of reference, axes fixed in space which coincide with 

 the principal axes through the centre of inertia of the body. Let 

 X, y, z be the co-ordinates of the point in contact with the rough sur- 



1 Cf. Jellett, "Theory of Friction," chaps, iv., v. 



