io Gwyther, Specification of Stress. 



possible cases, but the conditions with which I propose to deal 

 approximate in general character, though not in detail, to the 

 case of rest and motion of a rigid body, iand mjy object is to 

 consider the points of difference. 



I shall therefore assume that the components of the velo- 

 city of a particle may be written 



U= u B - u) s y + iOyZ + n , 



V— V — ttifZ + w s x + v , 



W= w - d) v x + w x y + w , 



where x, y, z 'are the coordinates pif a particle, that u , v , w ■ 

 b)„ io y , w„ are functions of / only, and that u, v, w are func- 

 tions of x, y, z, and /. 



We shall have 



and 



8& 8u . 

 — = — , etc. ; 



8JV + SV = ^ + ^ etc 



8y 8z 8y 8z 



8 IV 8 V , 8w 8v . 



— = 2w r + — - — , etc. : 



8y 8s 8v 8z 



and it will be assumed that the space-differential coefficient's 

 of u, v, w are small. 



Then on differentiation we shall find that the rate of change 



in U ; , ! I; !■''! I ;■ j | | 



= 8U + 8U U + 8U V+ 8U W 

 8t 8x 8y 8z 



in and similarly for the rates change of V and W. c Each of 

 these expressions consist of three parts ; 



1. A part independent of u, v, w. 



2. A part containing elements from u, v, w, and w A ., w y , w„. 



3. A part containing elements from u, v, w only. 



If the particles of a body are either constrained to move 

 or restrained from freje motion, they are subject to somd force, 

 and are in a state of stress. This is the case when a beam is 

 supported so as to prevent freedom of each particle of the beam 

 to fall under gravity, as: well as fin cases of motion, even, when 

 the material is supposed to be rigid. We may suppose the 

 number of (elements of the stress to be six, and that we have 

 not enough conditions drawn from the laws of motion to deter- 

 mine these elements. 



