470 REPORTS ON THE STATE OF SCIENCE. — 1919, 



We find 



Also 



EI 



) 



(5) 



If we now consider the bay immediately to the left, of which the end points 

 are, say, B' and A, and the length V and in which the end thrust is P', 

 we have in general a new n'. If we suppose that the cross-section of the 

 spar remains the same in each bay, we have, changing n into n' and 

 I into — V 



^l((}y\ =^_M / sin nT—7iT \_-^ /nT cos wT— sin nT\ 

 \dx}. ' "'Kiu/fV sin nTj \ {n'fl' sin n'V ) 



w 



- [2(1 -cos n'l')—l'n' sin n'l'] (6) 



2{ny sin n'l 



and since (5) and (6) must be the same, 



^ {&\n nl -- nl) -^ [sm n'V — n'V) 

 " nH sin nl '^ ""{n'fV sm n'V 



4-M ^""^ ^^^ ^^ — ^^^ nlj_a'V cos n'V — sin n'l'\ 

 ^ \ nH sin nl {n'fV sin n'V / 



4 W r ^(l~~^^Q ^ '*0 — ^'^ sin nZ , 2(1 — cos n'V) — Vn' sin n'l'~\ _„ 



L 2^' sin nl 2(^'f siSl?!' J ~ ^> 



which is the modified equation of the three moments. If we neglect 

 end thrusts, that is make P and P' approach zero and therefore also n and 

 n' approach zero, then (7) approaches the limit 



M3+Z'M,,+2M,(Z+r)='^ {I'-^l") 



which is the theorem of three moments. (See Love's ' Elasticity,' p. 381.) 



This modified theorem of three moments, together with the thrusts 



obtained by the ordinary theory of pin-jointed frameworks, enables the 



