2 Proceedings of the Royal Irish Academy. 



Take any rectangular axes fixed in space as axes of reference, and 

 let M, V, w be the components of tlie velocity of the point in the body 

 coinciding with the oiigin, and w„ Wy, w^, the components of the 

 angular velocity of the body. Also let 8 prefixed to any quantity 

 denote its increment produced in the small interval of time ht. Then 

 if X, y, % are the co-ordinates of any point in the body, we have 



X = u — yuz + zwy, 



p = V - ZWx + XWz, 



z = w- xuy + ywx. 

 Therefore at the time 8^ from rest 



5i; = S^< - ydooz + z^wy \ 



dy = dv - 2S«« + x^wz > , (1) 



Si = div — xSwj, + ySwx / 



for if 8^ be considered as a small quantity of the first order, Sx, 8y, Ss 

 are small quantities of the second order, since the body starts from 

 rest, and consequently terms of the type 8y . coj are of the thii'd order 

 of small quantities and may be neglected. 



If X, Y, Z, Z, M, iV are the constituents of the system of applied 

 forces referred to the axes, we have, by Newton's Second Law, com- 

 bined with D'Alembert's principle, 



XSt = S.dm . 5x \ 



= m (Sii — i/Saiz + zSaiy) I 



K (2) 



ZSt = 'Zdm [j/{S'W - xdwy + ydwx) - z{Sv — zSux + xSuz)^ 



= m {ySw — zSv) + aSoJx — hSaiy — ySaii J 



with similar values for YSt, ZSt, MU, NU, where x, y, z are the co- 

 ordinates of the centre of inertia, a, h, c, f, g, h them oments and 

 products of inertia, m the mass of the body, and dm an element of 

 mass. 



These equations hold even if the forces are not constant, but are 

 continuous functions of any variables, if we write for the constituents 

 of the force system their values at the instant of starting : provided 

 the variables are such that during the time ht they only change at 

 most by quantities of the same order of small quantities as St. For 

 then during the time St the forces would only vary fi'om their initial 

 values by quantities of the order St which would only introduce into 

 our equations quantities of the order {Sty which may be neglected. 

 The co-ordinates and components of velocity of any point of the body 

 are variables of this description. 



