CHAP. HI.] SUPPLEMENT TO CHAP. VII. 122 



From these two equations, and the two equations above, viz., V t f- V 2 = P 

 and M! + Vj li Mo V a Z 2 , we can determine Vi, V a , Mi and M a . 

 Thus from the last two we have 



Vi Zj+Va Z a = Ma-Mj = V li+P Z a +.Vi la = Vi l + P Z a , 



or V, 1= Ma-Mi-P Za. 



So also Va I = M 3 -Mj + P Zi, and substituting these in the equations above, 



we have 



Mi Z (2 Z!-Za)-Ma Z (2 Za-Zi) = P li Za (Z - Za) J 



and from these we have, finally, 



and then from the values of Vi Z and V a Z above 



> 



Change of shape. 



Substituting these values, we can now find 



=^l ['-( * + )] 



2 Z Z 



Hence y is a maximum for x ^-, and the maximum deflection 



3 li + li 



itself is 



2 P Zx Z a * 



A = 



This expression will be itself a maximum for l\ = Za or l\ = Z, that is, 

 the maximum deflection for a weight in the middle is at the weight and 

 equal to 



PZ 3 



A = 



192 B I ' 



This deflection is greater than the maximum deflection for any other 

 position of the weight, which in general is not found at the weight itself, 

 but at some other point between the weight and farthest end. 



We see above that the deflection in this case for load in middle is only 

 one-fourth as much as for same beam and load when supported at the ends. 



Breaking weight. 



For the greatest moment, which we easily find to be at the end, we have 



FZ.Z,' 



JYI 



Z* Z 2 



This is a maximum for Zi = i Z. That is, the greatest moment at the end 

 occurs when the load is distant one-third of the length from that end. The 



