AND THIN ELASTIC RODS. " 371 



employed. Of course when the displacement of the bow is considerable e is not constant, but 

 it is nearly so. 



The depression of the centre of gravity of AP is ^m, and EP = 4af. 



Hence if R be the upward reaction at B, »m, the mass of JB, nii of APy and m^ of half 

 the arrow, and if ZF be the force applied at P to stretch the bow, and R be assumed to 

 vary as w, and 5 be the extreme value of ED when the bow is fully bent, before the string is 

 released, we shall have by first principles 



^ d?ai F.V 



(mie + I nig + 4mz) -^ = ~ f 



Hence if V be the final maximum velocity of the arrow 



V- 



-V- ^* 



5 

 m^e + - mj, + 4»»3 



Ex. The unit of weight being 1 ounce, the accelerating force of gravity 32.2 feet per 

 second, to find the velocity when F = 200 x 16^.9, 



gm^e = 5.5 gnii = 1, 



gm^ = .5 and ^ = 1, 



r = 4\/ 



200 + 4 + 32.2 



5.5+2.5 + 2 ; ■ 



= 200 nearly. 



The greatest velocity I obtained was, I believe, about 215 feet, or, at the outside, 220 feet 

 a second with a bow, each leg of which was 20 inches long ; the stock in which it was set 

 2 inches broad ; the breadth of the bow 1 inch, its thickness near the stock 5- inch, and ^ inch 

 at the ends ; the initial value of BE was 4 inches, and of PE 12 inches ; ^ = 3 inches nearly. 

 The steel was of the best " St Etienne," forged and worked with great care. A bow made of 

 English steel, exactly similar, broke after using it some time, so I presume the steel is strairted 

 nearly to its breaking point, and F = 200 lb. probably. I may observe by the way that the 

 strength of steel varies greatly, and that the best fine-grained steel is not nearly so well 

 adapted for springs as a tougher and more irony steel. The velocities of the arrows I 

 measured by means of sights ; the depression for known short distances, combined with the 

 range at 45", enabled me very closely to calculate the velocity. The range at 45** gave the 

 resistance of the air, which was sufficient to reduce in some instances the velocity 10 feet a 

 second at 40 yards. A rough attempt at a Balistif Pendulum, which I constructed, invariably 

 gave the velocity too little, sometimes by as much as 20 feet a second. 



Let F = 80, or 9.F = 16O, ^ = -^ foot, and the bow be of wood. An ancient English 

 archer's bow probably "drew" as much as l6olb,, as even amateur archers use bows up to 

 100 lb. strength. 



