2 THE EQUATIONS OF MOTION. [CHAP. I 



bounded by a surface of revolution about this axis, would be 

 constant. We infer, moreover, that these tangential stresses are 

 not called into play so long as the fluid moves as a solid body, but 

 only whilst a change of shape of some portion of the mass is going 

 on, and that their tendency is to oppose this change of shape. 



2. It is usual, however, in the first instance to neglect the 

 tangential stresses altogether. Their effect is in many practical 

 cases small, and independently of this, it is convenient to divide 

 the not inconsiderable difficulties of our subject by investigating 

 first the effects of purely normal stress. The further consideration 

 of the laws of tangential stress is accordingly deferred till 

 Chapter XL 



If the stress exerted across any small plane area situate at a 

 point P of the fluid be wholly 

 normal, its intensity (per unit 

 area) is the same for all aspects 

 of the plane. The following proof 

 of this theorem is given here for 

 purposes of reference. Through 

 P draw three straight lines PA, 

 PB, PC mutually at right angles, 

 and let a plane whose direction- 

 cosines relatively to these lines 

 are I, m, n, passing infinitely close to P, meet them in A, B, C. 

 Let p, p 1} p 2 , p 3 denote the intensities of the stresses* across the 

 faces ABC, PBG, PGA, PAB, respectively, of the tetrahedron 

 PABC. If A be the area of the first-mentioned face, the areas 

 of the others are in order /A, wA, r&A. Hence if we form the 

 equation of motion of the tetrahedron parallel to PA we have 

 PI . ZA = pi . A, where we have omitted the terms which express 

 the rate of change of momentum, and the component of the 

 extraneous forces, because they are ultimately proportional to the 

 mass of the tetrahedron, and therefore of the third order of 

 small linear quantities, whilst the terms retained are of the second. 

 We have then, ultimately, p=pi, and similarly p=p 2 = p s , which 

 proves the theorem. 



* Reckoned positive when pressures, negative when tensions. Most fluids are, 

 however, incapable under ordinary conditions of supporting more than an ex 

 ceedingly slight degree of tension, so that p is nearly always positive. 



