STRAINED ELASTIC SOLIDS 473 



an, ... ass are all constant and given in value throughout the 

 solid. The same is true of the first minors ^u, . . . ^33. In con- 

 sequence [382a] combined with 



form a system of four equations to determine four "unknowns" 

 a, /3', 7', p, which will thus yield not only definite values of the 

 fluid pressure, but also definite orientations of the solid surface 

 compatible with this assigned state of strain. To see how 

 many definite values and orientations are involved we consider 

 [382a] carefully. Suppose that a definite value is assigned to p ; 

 this would give us three simultaneous equations to determine 

 the values of the unknown a', /3', 7', at least apparently. In 

 reality, however, we should have three equations to determine 

 two unknowns, viz., a/j' and 13' /y'. In short we have one 

 equation too many; values of a'/y' and jS'/t' which v»^ould 

 satisfy the first two would not necessarily satisfy the third, 

 unless a special relation existed between the nine coefficients. 

 The relation embodies the fact that the determinant of the 

 nine coefficients is zero, i.e., 



Xx' + Aiip Xy' + A12P Xz' + Anp 

 Yx' + A21P Yy> + A22P Yz' + Aizp 

 Zx' + A31P Zy' + A32P Zz' + Azzp 



= 0. 



Without actually multiplying this out, the reader will realize 

 that the left-hand side is an expression involving p, p^ and p^. 

 The equation is a cubic in p. Hence there are only three 

 values of p which are compatible with the state of strain. They 

 are the roots pi, p^, pz of this equation. If we insert one of 

 these values, say pi, into the first two of [382a] we can solve for 

 the ratios a'/y', ^' /y', and combining these with a'^ + /8'^ + 7'^ 

 = 1, we obtain values of a, /3', 7', say a/, /S/, 7/. Actually, 

 as is obvious, —a/, — jS/, —7/ will also satisfy the equations. 

 (Not of course —a/, jS/, 7/ nor any triad with an arrange- 

 ment of signs other than the two mentioned; for these would 

 give ratios not satisfying [382a].) Inserting p^ and pz we find 



