THE EFFECTS OF MACHINES WHEN IN MOTION. 153 



and heads are suspended in the common center of 

 gravity, P— ^r will be the moving power, and P 

 ^J^^^-f; the sum of all the mass after being 



llVf' 



in motion. Hence P + ^ + ^1"+^^ is the accelera- 



tive, and P +^^+-^^+/ the motive force of d\ 

 equal —^, by putting t-'?-\-''-^V~, and by tak- 

 ing its fluxion equal nothing, j;z:L\/f+tV-^i. 



Prob. 17. To determine .v under the circum- 

 stances of a maximum, when the two circular ends 

 are braced to the main beam ab, by the braces v, 

 "whose weight altogether is ^zv . 



Then, by Problem 11, the// of two of the "os at 



one end, reduced to B, is ,„z,l — u . , ^4-— ^~ 



-^ by putting /c^—-=:=^y—^^^, and where^e:^^. 

 Hence the p' for all the braces 

 reduced to B, will be ^-^, and 



•" R^ ' 



by the same problem— -f-r 

 ^-^ will be thep' of the whole 



beam, heads and braces, re- 

 duced to B. Now since the 

 ends are in equilibi'io, exclusive 

 of the weight, P— .r will be the 



moving power, and P+ — + —z-\ — — ^s the motive 



force of .V, which by putting P+'^4-^+-^~^' 



becomes ^^^ , and by making its fluxion equal to 



nothing, we shall have .v — \/t^-{-tP — f, as in the 

 former case. And this form will always obtain for 

 all beams moving on their centers, after determin- 

 ing the value of/, and substituting ^ for the known 

 terms in the denomination. 



PflOB. 18. 



