428 RICE ART. K 



It must not be forgotten that this analysis relates to any 

 arbitrary choice of axes of reference. Actually it is possible, 

 by selecting a special triad of orthogonal lines as axes, to intro- 

 duce a diminution in the number of stress-constituents required 

 for the formulation of the stress across any given plane at a 

 given point. A proof of this statement appears in Gibbs, I, 194, 

 195, but it is not so famihar and not so easy to grasp as the 

 usual proof given in works on elasticity, which follows a line 

 of reasoning similar to that adopted earlier to indicate the 

 existence of three principal axes of strain, and is here outlined. 



Conceive that a quadric surface whose equation is 



is constructed with P as center and with any local axes of ref- 

 erence at P; Xx, Xy, . . . Zz being the values of the stress con- 

 stituents at the point P. Let a line whose direction cosines 

 are a, 13, 7 be drawn from P cutting this quadric in the point 

 Q; denote the length of PQ by r so that the local coordinates of 

 Q are ra, r^, ry. Now draw the tangent plane at Q to the 

 quadric surface and drop PN perpendicular to this plane. By 

 the theorem already used we know that the equation of this 

 tangent plane is 



(Xxra + AVr/3 -f X^ry)^ + {Y^ra + YyVlS -^ Yzry)r, 

 -f (Z^ra + ZyrlS + Z^ry)^ = k 



(remembering that Yz = Zy, etc.), and so the direction cosines 

 of PA'' are proportional to 



aXx + fiXy + yXz, aYx + ^Yy + yY z, 



O^Zx + (3Zy + yZz. 



Thus a glance at (28) shows us that the stress action at P across 

 a plane normal to PQ is itself parallel to PN. In general PN 

 is not coincident with PQ, i.e., the stress action across any plane 

 is in general not normal to the plane, as we know already; but 

 the information now before us about its direction indicates that 

 there are three special orientations of the plane for which this 

 happens to be true and for which PA^ lies along PQ. They are 



