250 SUPPLEMENT TO CHAP. XIH. [ART. 7 



In this case we have from (9), since now M = 0, 

 H, = -?*!+P(l-*), K, = ^l + ^=-Pk = Pk, R, = -^, 



M *1 1 *9 



and R 4 = 0. 

 By the conditions R must equal R,, hence 



2 Z, M a + Z, M, = - Z, I, P Te ...... (16; 



From the equation of three moments above, we have, making r 1 = 8, 

 or r = 8, since then P disappears, 



M a ?, = -, ........ (17) 



01 M, = - ^-'. 



it 



Substituting this value of M a in (16), we find 



T, = l ^2 , hence M, = -^, 

 2 Zi + 2i 2 Zi + Z a 



and therefore, at once, 



Putting n I in place of Z a , we have 



R, + 2 R a , it will be observed, equals P, as should be. 



The conception of a beam tipping, as in the last two Arts., is due to Clem- 

 ens Herschel (Continuous, Revolving Drawbridges, Boston, 1875), and the 

 above formulae were first deduced by him in the above work. 



