Apeil 29, 1910] 



SCIENCE 



675 



dent reports the rest take notes, just as they 

 do when the instructor lectures. At the end 

 of each report questions are asked and correc- 

 tions are made. The notes taken hy the rest 

 of the students are corrected by the one who 

 gives the report, and are bound up with the 

 students' general note-book for the course. 

 The one reporting binds up his outline, and a 

 list of the books and papers consulted — a bib- 

 liography. 



By this plan the student learns much about 

 one animal not treated in the texts and he 

 learns a little about a good many other 

 species. But he does more — he gets a training 

 in using the powers of observation, in order- 

 ing the facts oitained and in expressing to 

 others the knowledge gained. 



The two main suggestions are worth a trial 

 by other teachers. The university should en- 

 courage the teaching of zoology by becoming 

 a center for furnishing and distributing the 

 material for the preparatory schools of a state 

 at cost. Much of this could be secured very 

 cheaply by a collecting expedition to Puget 

 Sound. The student should be given the prob- 

 lem of furnishing the rest of the class with a 

 report dealing with a special form of animal 

 life somewhat closely related to a type studied. 

 This working out of a " lecture ?" by the stu- 

 dent is the best of training for him. 



W. J. Baumgartner 



SPECIAL ARTICLES 



AN EXPRESSION FOR THE BENDING MOMENT AT 



ANY SUPPORT OF A CONTINUOUS GIRDER 



FOR ANY NUMBER OF EQUAL SPANS 



Tables giving the bending moments at the 

 supports of a continuous uniformly loaded 

 girder with equal spans are found in most of 

 the books on strength of materials, but these 

 tables usually stop at six or seven spans. The 

 object of this paper is to give a general ex- 

 pression from which the bending moment at 

 any support for any number of spans can be 

 computed. First the expression and explana- 

 tion of the method of computation are given 

 and then follows the derivation of the formula. 



Let J/„ -Mn, • ■ • be the bending moments at 

 the first, second . . . support, respectively. Let 



n be the number of spans, w the load per 

 unit length and I the length of span. If Mr 

 represents the bending moment at the rth 

 support then the formula gives 



Ar_,Z)„_rrt — Z)r-=A„-r ,, 



Mr — —- 



2A„.i 



The As and Ds are numbers to be com- 

 puted from the formulas. 



A,t = 4A„.j — A,,.,, 

 Z)„ = A„.i — /)„_!. 



As shown below, A„ = 1, A^ = 4 and i?„ = 

 and any other A or D may be easily computed. 

 For example, 



A2 = 4Ai — A„=15, 



A3 = 4A2 — A, = 56, 



D, = Ao — Do = 1, 

 Z), = Ai — Z), = 3. 



Thus, if, for example, we wish the bending 

 moment at the fourth support for seven spans, 

 we have r ^ 4, n = 7 and 



A;A — D.,A, 

 2A, 



wP. 



From the above formulas A^ = 15, D^ = 4A, 

 D, = 3, A3 = 56, Ae = 2911. Hence 



[M.], span. =—6/71 WZ=, 



a result which is verified by the tables. 



The derivation of the above formula is noth- 

 ing but the general solution of the equations of 

 three moments by determinants. For n spans 

 we have, from the theorem of three moments. 

 Ml + iM. -t- M3 = — wV/2, 

 M^ + iM, + Mt = — wiV2, 



if „-i + 4il/„ + i)/„« = — icl-/2. 

 Since M^ = M^^^ ^ we have left n — 1 equa- 

 tions with n — 1 unknowns. If we write 1 

 in place of — wr/2 and multiply the final re- 

 sult by — it>P/2 the solution will be less com- 

 plicated. Writing the Ms with the same sub- 

 scripts under one another we have 



4K, + M3 =1, 



M2 + 4:Ms + M^ =1, 



M3 + 4Mi -}- M5 = 1, 



The determinant of the system of equations 

 will be the determinant. 



