226 



PRACTICAL MATHEMATICS 



OP and OQ be another pair of rectangular axes inclined to the 

 first pair at an angle a. 



If G represents the position of an elementary area a, and the 

 co-ordinates of G are x and y with reference to the axes OX and OY, 

 then the moment of inertia of the lamina about OX = 2ag/ 2 = X, 

 and also I OY = Sao: 2 = Y 



FIG. 58. 

 With reference to the axes OP and OQ, 



-Inp = itCL (jrj_i = Jr 



and 

 Now 



I OQ - 



Sa GH 2 = Q 

 GL = GM - ML 

 = GM - KN 



= y cos a x sin a 



GH = OK+KL 

 -OK+NM 



= x cos a + y sin a 

 GL 2 = (y cos a x sin a) 2 



= y 2 cos 2 a + x 2 sin 2 a 2 any sin a cos a 



= y 2 cos 2 a -f x 2 sin 2 a xy sin 2a 

 GH 2 = (x cos a + y sin a) 2 



= x 2 cos 2 a + ?/ 2 sin 2 &.+ xy sin 2a 



P = Sa GL 2 = cos 2 aZa/ 2 + sin 2 aiEax 2 - 



= X cos 2 a + Y sin 2 a - Z sin 2a 

 where Z = Itoxy 



Q = Sa GH 2 = sin 2 aSm/ 2 + cos 2 aSa# 2 + sin ZtxZaxy 

 = X sin 2 a + Y cos 2 a + Z sin 2a 



Then P - Q = X(cos 2 a - sin 2 a) - Y(cos 2 a - sin 2 a) - 2Z sin 2a 

 = (X - Y) cos 2a - 2Z sin 2a 



and 



Then 



Also 



Hence 



Also 



