CHAP. XIH.] ANALYTICAL FORMULA. 221 



us take the example of Art. 127 (Fig. 88), where P = 40 tons, 

 =80 ft., and a becomes 10, 30, 50 and 70 ft. respectively. 

 First, as regards the expressions A A' : 



These become in the present case 3200 (2 k 3 J& + JP) and 

 3200 (k ^) respectively, where k has the values -, -f, -|, and -J 

 successively. Now as the denominator is in each term always 

 the same, in the first 8, in the second 64, in the third 512, and 

 only the numerators of the values of k vary for the different 

 positions of P, we may put these values of A and A' in the 

 forms 



or 



A = 800 A 150 A 2 + 6.2305 A 3 , 



A! = 400 h - 6.2305 A 3 , - 



where A has successively the values 1, 3, 5 and 7, for Pj, PS, 

 P 3 and P 4 respectively. These are then the practical formulae 

 for substitution in the present case. 



"We can now apply the formulae for M above. Thus, sup- 

 pose for seven spans we have P 3 in the fourth, as shown by 

 Fig. 88, and wish the moment due to P 3 at the fourth support 

 D. Then s = 7, r = 4, and m = 4, and we have 



A<? 5 + A'<? 4 

 JYE 4 = C 4 , 



or, referring to Art. 136 for the Clapeyronian numbers, 

 -56 A + 15 A' 840 A -225 A' 



M 4 = - 15 



2911 2911 



5 



Now for P 3 we have & = -, or h = 5, and therefore 



8 



A = 1028.81 A' = 1221.2. 

 Hence 



589430.4 000 , f . 

 M * = on n = 202.4 ft. tons. 

 .zyii 



This divided by 10 = height of truss gives tension in A a = 

 20.2 tons, nearly what we have already found in our tabulation, 

 Art. 127. 



In like manner we may easily find the moment at D due to 



