CHAP. HI.] SUPPLEMENT TO CHAP. VII. 119 



P I * 

 have O' = O 4- ' . We have then the two equations 



/3 E X 



P Zi 



containing both the same constant O. 

 Integrating these, we have 



P Z a x 3 P Z, rl x* &~i P 



- 



In the first of these, for x = 0, y = ; hence d = 0. 



P h A 

 For x = Zt, y = y 1 ; and hence O = -- . 



6 E I i 



P Z Z 



For * = Z, y' = ; and hence, finally, O = ' | (2 Z Zi). 



6 E It 



We have, therefore, by substitution of these constants, 



p 7 2 72 

 For x Z,, we have the deflection at the load y = -. 



O E I 6 



Inserting the value of O in the value for -J- above, and placing the 



a x 



value of - equal to 0, we have for the value of x, which makes y a maxi- 



d x 



mum, * = / (2 Z Zj) Z l5 an expression holding good only for x less than 



Zi. Inserting this in the value for y, we have for the maximum deflection 

 itself 



If the load is in the middle, we have for the curve of deflection 



* P Z 3 



and for the deflection itself A = 



48 El' 



The greatest deflection is not, then, at the weight, except when the load! Is 

 in the middle. When this is the case, the deflection is only -jVth of the 

 deflection for the same length of beam fixed at one end and loaded at the 

 other or free end. 



15. Beam as before supported at the ends Uniform 

 load. For a load p per unit of length, the entire load is p I. The reac- 



tions at each end are ?, and the moment at any point is 



M is evidently greatest at the centre, and hence 

 Max. M = -^!. 



