456 



Mr. A. Lodge on the Relations between 



The proof is the converse of the 

 preceding proof. The only thing 

 requiring special consideration is 

 whether the axis of minimum 

 moment will fall in the proper 

 quadrant KOL, i. e. in the positive 

 quadrant if OM is positive, and 

 out of it if OM is negative. 



The axis of minimum moment 

 will evidently be in the same quad- 

 rant as the point U from which 

 the moments are measured, and 

 by the construction U will be in 

 the first quadrant if OM is posi- 

 tive and in the second if OM is 

 negative ; which completes the 

 proof. 



The expressions for the principal moments in terms of the 

 given moments and product are easily deduced from the figure, 

 viz.: — 



a 2 , 6 2 =UT + TA=UT+ VTM 2 -fMK 2 





Fig 



3. 





U 





B 



/ 



+ L, 



/ / 



/ 

 / 

 / 

 / 

 / 

 / 

 / 

 / 

 / 



T j 











A 



v 







V 



= i(F + Z 2 )+i V(& 2 -Z 2 ) 2 + 4A 2 , (4) 



and the angle AOK (0) is given by the equation 



n 



tan 20= tan KTA: 



P-l 2 



(5) 



These equations are of course deducible from the relations 

 (1), (2), and (3). 



Stresses and Strains. 



The above construction is also very useful for the graphic 

 determination of principal stresses and principal strains from 

 given two-dimensional stress or strain conditions in any two 

 rectangular directions, as the equations are exactly of the 

 same form as (4) and (5). 



Thus, let OX, OY be two given rectangular directions, and 

 let^ be the normal component and q the tangential compo- 

 nent of a given stress on planes perpendicular to OY, and 

 p 2 , q the normal and tangential components of a given stress 

 on planes perpendicular to OX, the whole action being 

 restricted to the plane XOY; and let p 1 , p 2 be considered 

 positive when tensile, and q positive when it tends to produce 

 positive sliding, i. e. when it tends to diminish the angle 

 between + OX and + OY. 



