164] HOMOGENEOUS STRAIN. 429 



Let us examine the conditions that the two vectors OP and OP' 

 shall have the same direction. The condition for this is 



where A is to be determined. Introducing the values x' = &x, y' = ky, 

 % ] = fig into equations 1) we obtain 



(% A) x 4- a 2 y 4- a 3 # = 0, 



4) \x + &-A)y 4-M = 0, 



c^ + c 2 # 4- (c s A) = 0, 



a set of linear equations to determine x, y, z. The condition that 

 these shall be compatible is that the determinant of the coefficients 

 vanishes. 



5) 



, 6, 



0. 



This is a cubic in A. Let its roots be A lt A 2 , A 3 . Inserting any 

 one of these in 4) we may find the ratios of x, y, z giving the 

 direction of the vectors in question. 



Supposing that A 1; A 2; A 3 are real, let us find the condition that 

 the three directions are mutually perpendicular. 



Substituting first I = ^ and then I = A 2 in 1) and 3), we have, 

 denoting the values of x, y, z by corresponding subscripts, 



+ Vi^ 

 i + & 2/i -+ ^^ = 

 ^i + 



-f 



Multiplying the first three respectively by # 2 ,2/ 2 ,'# 2 and adding, and 

 subtracting the sum of the last three multiplied respectively by 



x # we 



7 ) fe - & 8 ) (2/1^2 - *i 2/ 2 ) 4- (OB ~ c i) (^i ^2 - %^ 2 ) 4- (&i- a 

 = (A! - A 2 ) fe^ + 2/^ 2 4- ^^ 2 ). 



The condition for perpendicularity of r lf r 2 is 



% 4- // + ^^ = 0. 



