CHAP. Xin.] ANALYTICAL FOEMUL.E. 202 



and for all kinds of regularly and irregularly distributed and 

 concentrated loads both for constant and varying cross-section 

 of girder. His formulae are mathematically exact, and for 

 given loading are free from integrals. 



The above is but a very imperfect sketch, and we have named 

 but a few of the many writers who have been occupied with 

 the subject. Olapeyrori' 's Theorem above alluded to, as origi- 

 nally given by him, applied only to uniform load over whole 

 length of girder, or over an entire span. But as early as Bresse's 

 Treatise, it had been extended to include concentrated and 

 local loads as well, and WinJder has also given a very complete 

 and practical discussion of the subject. 



Notwithstanding the labors of these and many other mathe- 

 maticians, there seems to be a wide-spread idea, even among' 

 those who are supposed to have considerable familiarity with 

 mathematical literature, that the results deduced are unpracti- 

 cal. It is not uncommon to meet with even recent publica- 

 tions* in which it is stated that the authorities pass over such 

 problems with "judicious silence;" that the mathematical in- 

 vestigations are intricate, and the formulae deduced trouble- 

 some in application ; that even a " partial solution of the prob- 

 lem by mathematical calculation is attended with considerable 

 difficulty, and that a complete solution for the bending moment 

 and shearing force at every section, under moving partial and 

 irregular loads, taxes the powers of the best mathematicians, 

 and is well-nigh impossible, so far as any practical application 

 of them by the engineer is concerned." How far such ideas 

 are justified may be seen from the following pages. That the 

 authors and works above referred to can only be read by good 

 mathematicians is not to be denied. It may also be admitted 

 that the subject is an intricate one, and when treated mathe- 

 matically in its most general form the results are naturally in an 

 unpractical shape. But that these results are, therefore, worth- 

 less, or that the formulae, when applied to any particular case, 

 are " too intricate for practical use," by no means follows. 



The desirability of formulae for the application of our graph- 

 ical method as developed in the preceding chapter ; the erro- 

 neous ideas prevalent on the subject which we have just noticed ; 



* Graphical Method for the Analysis of Bridge Trusses : Greene. D. Van 

 Nostrand, publisher, New York, 1875. 



