CHAP. Xm.] ANALYTICAL FORMULA. 229 



For concentrated load, 



P T 2 f 9 & 9 2 -L Z-^ A f P T 3 /7- 1P\ 



f r \ KJ On -p A/ I, .tX ^ Jr t- r ^/l/ ~ K> I, 



and ^ = P (1 ^), ^' = P ^j ^ being always y. 

 The quantities denoted by are also as follows : 



ff (4: + 3jp) + n (4 + 



__ - 



P 



_ p (14 + I2p)-n (14 + 



P 



_ p (52 + 45 p] + n (52 + 60 p] 



P 

 -p (194 + 168 ff) - n (194 -f- 224 p) 



p (724 + 617 p} + n (724 -f 836 p} 

 Co ' . etc.. 



P 

 following the law of the Clapeyronian numbers. 



151. Application of the above Formulae. The for- 

 mulae of the preceding Art. comprise in a most compact form all 

 the formulae hitherto given, and are all that is necessary for the 

 complete solution of any practical case. 



Thus, by making p = unity and retaining only n, we have 

 the case of a girder with variable end spans n I, of different 

 length from the others, which latter are all equal and repre- 

 sented by I. The reader will find no difficulty in using the 

 above. For any particular case, when w or P and I, k, n and 

 p are given. A, A', q and q' can be easily found, and the prob- 

 lem is solved. If n and p be both unity, we have the formulae 

 for all spans equal. The expressions for M,,, will then reduce 

 to those already given in Art. 144. Thus, in Art. 145 we have 

 already found for seven equal spans, I = 80 ft, load P = 40 

 distant 50 ft. from left ; the moment M 4 = 202.4. Now, from 

 our formulae above, we find for Mg making m = 5, 5 = 7, 



Then by our formulas for shear, S 4 = + 14.12, or nearly 



