STRAINED ELASTIC SOLIDS 483 



minimum conditions of a function of several variables, and 

 resembles that employed by him on pages 194, 195 when 

 demonstrating the existence of the principal axes of stress. It 

 will be followed easily by those versed in such analytical 

 methods, but for other readers not so well acquainted with 

 mathematical technique we can give a geometrical flavor to the 

 argument which may prove helpful. We saw in the previous 

 discussion that 



is the equation of a locus drawn round the local origin P' which 

 is strained into a sphere around the center P. This locus is an 

 ellipsoid, and its actual form and the orientation of its principal 

 axes in the body are of course dependent entirely on the magni- 

 tude and nature of the strain and not at all on the particular 

 choice of the axes of reference, OX', OY', OZ'. We have already 

 seen in this article that the principal axes of this "elongation 

 ellipsoid" experience no shear and so are the principal axes of 

 strain, and we can therefore proceed at once to the deduction of 

 equations [430] and [431] on page 207. The method is well 

 known to students of analytical geometry. Suppose that R' 

 is a point in which one of the principal axes of this elongation 

 ellipsoid through its center P' cuts the surface, and let its local 

 coordinates be ^Z, tji', f/. We know that the direction cosines 

 of the normal at P' are proportional to 



But since P'R' is along a principal axis, the normal at R' coin- 

 cides with P'R' and so the direction cosines are also proportional 

 to ^i, r}i', fi'. Thus the three quantities 



fli^i' + aem' + ctBfi' fle^i' + a2Vi' + «4fi' 



; ' ; ' 



F~' ' 



