(1HAP. Xin.] ANALY'HCAL FORMULA. 211 



these equations can be solved by the ordinary rules of algebra ; 

 but for a great number, or in the general analytic discussion of 

 any number, we must have recourse to a special artifice. Thus 

 we multiply our equations, beginning with the last, by numbers 

 indicated by <? 15 63, c 3 , .... c^_i, and then choose these num- 

 bers such that, by the addition of all the equations, all the M's, 

 with the exception of M 1? disappear. We thus easily, determine 

 M L without the tedious process of substituting from one equa- 

 tion to the other, through the entire list. 



The following relations must then evidently hold between 

 these numbers, as is evident from the theorem of three mo- 

 ments of Art. 131 : 



2 c l (4_ t + 4) + <fe 4_! = o. 



Oi 4-1 + 2 <h (4-2 + 4~l) + C S 4_ 2 = 0. 



c s _ 3 4 + 2 c s _ 2 (4 + 4) + 8 -i 4 = o. 



If the first number is chosen at will, say 1> the other num- 

 bers can be found from these equations. 



Now in the present case of all spans equal, we have between 

 any three of these numbers the relation: 



<W-i + 4 G m + c m+1 = 0. 



If we take the first, ^ = 0, and the next, <% = 1, we have 

 for the others the following values : 



d = c 4 = + 15 ci=- 780 c 10 = + 40545 



63 = + 1 c 5 = - 56 <V= + 2911 c n = - 151316 



c 3 = 4 c 6 = + 209 c 9 = - 10864 c^ + 564719 



These are the so-called Clapeyronian numbers. They alter 

 nat*, as we see, in sign, and each is numerically 4 times the 

 preceding minus the one preceding that. We shall always indi- 

 cate these numbers by the letter c, the index denoting the par- 

 ticular one. Thus, c^ is the seventh number, counting and 

 1 as the two first. 



No table of these numbers is needed. The index being 

 given, any one can write down the series for himself, till he 

 arrives at the desired number. 



137. Uniform Live Load over any ingle pan Moments 

 at Supports of Loaded Span. These numbers being pre- 

 mised, we can now give the following formulae for the momenta 

 at the supports r and r 4- 1 of the uniformly loaded span : 



