STRAINED ELASTIC SOLIDS 419 



tinguished from the stresses which are occasioned by them. 

 To give a definition of the "stress at a point," we must conceive 

 a surface, on which the point Hes, dividing the body into two 

 parts. We also conceive a small element of this surface sur- 

 rounding this point. Of the total force which we imagine one 

 portion of the body to exert on the other across this surface, a 

 certain small part is considered to be exerted across this element 

 and, when the element is small enough in size, to be practically 

 proportional in magnitude to the area of the element and 

 unchanged in direction as the element is made smaller and 

 smaller. The quotient of this force by the area is assumed to 

 have a limiting value as both are indefinitely diminished in 

 magnitude. The reader is certainly acquainted with this con- 

 ception in the case of liquids and gases; but in such a case there 

 is a special simplification. For one thing the force is almost 

 always in the nature of a thrust in a fluid medium; in a solid 

 medium it may be a thrust or a pull. Moreover, in the case of a 

 fluid at rest, the force is normal to the element of the con- 

 ceptual surface. That is not in general the case for solid media. 

 The limiting value of the quotient of force by area referred to 

 above is called the stress across the surface at the point, and, as 

 stated, it is not as a rule directed along the normal to the 

 surface at the point. Another important distinction should be 

 noted here. In the case of a fluid not only is the pressure always 

 normal to the element, but it retains the same value as the 

 element assumes different orientations. (If the reader has 

 forgotten the proof of this it would do no harm if he refreshed 

 his memory, as the proof involves some considerations of value 

 to us presently). But in the case of a solid medium the 

 stress generally alters in value, as well as direction, as the 

 orientation of the element of surface is changed. In the 

 technical language of the vector calculus, the stress is a 

 vector function of the unit vector which is normal to the 

 element and changes in magnitude and direction as the unit 

 vector is turned to be in different directions. In the case of a 

 fluid medium at rest one numerical magnitude is obviously all 

 that is required to specffy the pressure at a point, and the physi- 

 cal problems raised involve the functional dependence of this pres- 



