CHAP. V.] MOMENT OF ROTATION PARALLEL FORCES. 55 



1. When a certain number of wheels pass over the truss, but 

 without any passing off, or new ones coming on ; what position 

 of the system gives the maximum moment at any given cross- 

 gection not covered by the system, and how great is this 

 moment ? 



2. Under the same supposition as above, what position of the 

 load gives the greatest moment for a given point covered by one 

 of the load systems ? 



3. Among all the various points of the span, at which is found 

 the greatest maximum moment, for what position of the load 

 does it occur, and how great is it ? 



4. If the number of wheels is indeterminate, how many mast 

 pass on, and what position must they have to give at any point 

 the greatest maximum moment ; where is the corresponding 

 cross-section, what position must the load have, and how great 

 is this maximum moment ? 



The three first questions are easily solved by the aid of the 

 above properties of the parabola, enveloped by the closing lines 

 of the equilibrium polygon, corresponding to different positions 

 of the span. 



Thus, as regards the first question, let the given cross-section 

 be 5, PI. 9, Fig. 32 (d), and suppose the span s s in the position 

 where the vertical through b intersects a- a- at the point of tan- 

 gency (3. When now the span shifts, the intersection of the 

 ordinate through 5, with the corresponding tie line, will always 

 lie upon a- a: But this ordinate gives the reduced moments for 

 (reduced to pole distance H.) The greatest of these moments 

 will then be simply the greatest of the ordinates between a- <r 

 and the polygon, and will always be found at an angle of the 

 same. When found, we have at once the position of 5, and of 

 course of the span with reference to the given loads. This is 

 always such that a wheel stands over the given section. 



Thus in Fig. 32 (a), supposing the four wheels P 6 to P 9 to 

 pass over the span t\ t^ we seek the position of the load to give 

 the greatest moment at a point $ f tne span from the left, 

 therefore ^th from the middle. 



We lay off the span in such a position, t v f l5 that its centre is 

 distant from the intersection a of the outer lines of the poly- 

 gon by Jth of the span. 



The ordinate through the given point now passes through the 

 point of tangency of the tie line and parabola. We draw this 



