. Eine Abhandlung über die mathematische Theorie der Elastizität . rdingly dy inAQ B^ = -^ F^dx i oe, - ^Q^ di = dy W i (l-x). 247] AUF EINER SPANNE 379 Wir integrieren diese in die Formen B (?/ - A; tan A) = - ^ Z- FF V,B{y-(l- A;)tan/3] = - i l-W^{l- xf, wobei tana und tan/3 die Abwärtshänge der Mittellinie an den Punkten A und B sind. Die Bedingungen für die Kontinuität von y und dyjdx bei Q sind B ; tan A - ^ Z-i TF f f = Bf tan /3 - ^ Z F^p, B tan A - ^ ?-i TTF f = - B tan JS + i Z TFF^=- Diese Gleichungen geben B tan A = i l-^^ W^ ( + 2r), B tan /3 = J Z-^ Wenn ^f (2? + f )? Daher in J.Q, wobei f > a; > 0,

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. A treatise on the mathematical theory of elasticity . rdingly dy inAQ B^ = -^ F| dx I oe, -^Q ^di = dy _ W^ I (l-x). 247] ON ONE SPAN 379 We integrate these in the forms B (?/ - a; tan a) = - ^ Z- Ff V, B{y-(l- a;)tan/3] = - i l-W^{l- xf, where tana and tan/3 are the downward slopes of the central-line at thepoints A and B. The conditions of continuity of y and dyjdx at Q are B |; tan a - ^ Z-i Tf f f = Bf tan /3 - ^ Z F^p, B tan a - ^ ?-i TTf f = - B tan jS + i Z TFf |=- These equations give B tan a = i l-^ W^^ (^ + 2r), B tan /3 = J Z-^ If ^f (2? + f )? Hence in J.Q, where f > a; > 0, we have B2/ = iZ-Fr|?(? + 2r)^f-*i, (5) and in J5Q, where l>x>^, we have By=ii-F^{r(2r + n(^-^)-(Z-^)n (6) We observe that the deflexion at any point P when the load is at Q is equal to thedeflexion at Q when the same load is at P. The central deflexion due to the weight of the beam, as determined by the solutionof case (6), is the same as that due to | of the weight concentrated at the middle ofthe span. (e) Concentrated load. Built-in ends.. Fig. 38. To the values of By given in (5) and (6) we have to add the value of By givenin (3), and determine the constants M(, and J/j by the conditions that dyjdx vanishesat i!;=0 and at ^•=i!. We find from which M^=W^^^ll M^=Wi^jl Hence in AQ, where |>a;>0, we have and in BQ, where l>x>^, we have ?By^l-^Wp{l-xY{Z^.v-^{l-x)}. 380 THE THEOREM OF [CH. XVII We notice that the deflexion at P when the load is at Q is the same as the deflexionat Q when the same load is at P. The points of inflexion are given by cPyldx^=Q, and we find that there is an inflexionat P^ in AQ where APi = AQ. ABI{ZAQ + Bq). In like manner there is an inflexion at P^ in BQ where BP^ = BQ.ABI{ZBQ+AQ). The point where the central-line is horizontal is given by dyldx=0. If such a pointis in AQ it must be at a distance from A equal to twice APi, and for this to happenAQ must be >BQ. Conversely, if AQ<BQ, the point is in BQ at a distance from

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