14 



Proceedings of the Royal Irish Academy. 



and if motion is not possible, that it has no negative root. Suppose 

 ft! is the coeificient of friction necessary to prevent the point zy% 

 moving ioitially. Then, since ia this case Smj = 0, hcz = 0, vfe have 



-Pi ^^12 A. 13 



2 Pi l+Au 



A13 



2 



Pi 



1 + Aii 



A12 



P2 1 + A.22 A.23 



= -P2 A.12 A23 



+ 



P2 



A12 



L+A22 



-P3 A23 1 + A.33 



Pa Ai3 1 + A33 





Pa 



Aia 



A33 



efore equation (9) can be written 









Pi A12 



Al3 



2 



^2Pi2Z;4 + PyL-3 + ^^-2 + p^- + (^2 _ ^'2) 



P2 1 + A22 



A23 



= 0, 







Pa 



A2 



i 



1 + A33 





when, for convenience, we have written P, Q, and P for the coeffi- 

 cients of l^, Jc^, h, respectively. 



Consequently, U. fjL> fj/, the equation has an even number of nega- 

 tive roots or none, and if yu, < /x', an odd number. 



As a particular example we will consider the case where the normal 

 to the rough surface at the point of constraint passes thi'ough the 

 centre of inertia of the body. Take rectangular axes through the 

 centre of inertia, such that the axis of x coincides with the normal, 

 and for the present as the axes of y and z any two lines consistent with 

 this. 



Let X be the distance of the constrained point from the origin, 

 and denote by Pi, Pz, P3 the component of the initial reaction parallel 

 to the axes. 



Our dynamical equations are 



m5u ={X + Pi) St, mSv = ( F + P2) df, mSw = {Z+ P3) dt, (10) 



aSux — hSccy — gSaii = ZSt, 



- Mwx + iSwy - fSojz = [M - xRz) St, (11) 



- gSuix -fSwy + cZoiz = {N+ xSi) St. 



The geometrical equations are 



5m = 0, 

 m {Sv + xSco.) = kE2St, (12) 



m {Sw - rfScoy) = kSzSt, 

 where Jc has the same meaning as before. 



