58 MOMENT OF ROTATION PARALLEL FORCES. [CHA1. V. 



moment given by the ordinate at 4 is for either case almost 

 exactly the same. Any less span including the four weights 

 would give a less moment ; less, therefore, than the moment 

 already caused by the three weights. The span s' 2 s' 2 then 

 precisely as before, is the extreme limit upon which the 

 three wheels P 8 to P 6 cause the greatest possible maximum 

 moment. 



In a precisely similar manner we find that the span $' 8 s' s 

 with a centre midway between 03 and 4 is the limiting span for 

 the four wheels P 2 to P 5 . 



If now the span still increases so that PI comes on, the inter- 

 section of the outer polygon sides falls in our Fig. nearly at s ly 

 and since this point also happens to correspond almost exactly 

 with the angle 3, we take the centre of the beam at s t . The 

 greater the span now becomes, the greater the maximum 

 moment. The greatest length, however, which the span can 

 have without including P 6 , is twice 81 6, or twice the distance 

 between s x and P 6 . If P 6 also comes on, the intersection of the 

 polygon sides is found at a 5 , and the nearest polygon angle is 4. 

 Midway then between a 5 and 4 is the new centre of the beam, 

 while before P 6 came on, it was nearly at s t . But for centre 

 ! the half span was Si 6, while now it is somewhat less than 

 4 6 ; therefore considerably smaller. Since, however, we wish 

 to follow the span as it continues increasing, we must compare 

 those two spans which are equal before and after the coming 

 on of P 6 . The right-hand ends of these spans, viz., s' t and s s 

 must evidently be distant each side of 6, by the half distance 

 of their centres t and 4, or 0% (more accurately the point half- 

 way between a^ and 4, but 05 and 4 lie in our Fig. so nearly to- 

 gether that the centre cannot be indicated more exactly). We 

 make then 81 ' 4 = a% s s = M t 6, provided that M x is taken half- 

 way between the centre $ : and 0%. 



An exact construction shows that the maximum moments for 

 these two spans, the one given by the ordinate through 3, the 

 other by the ordinate through 4, are almost exactly equal, and 

 moreover, that the maximum moment for the span B! B! of equal 

 length whose centre is at ML is also almost exactly equal, when 

 measured upon the vertical through M^ We can therefore 

 take Sj Si as the limit of those spans for which the five wheels 

 P t to P 5 cause the greatest maximum moment. 



Taking on now the seventh wheel, the intersection of the 



