372 Mr ROHRS, on THE MOTION OF BEAMS 



If the denominator of the formula = 5, which would be somewhat near its value, the 

 whole string weighing about ^ oz , and the arrow 1 ounce, V = 250 nearly. I suppose that 

 l60 lb. would be, however, quite the limit of the strength of an archer's bow under any cir- 

 cumstances. V probably never exceeded SOO feet a second, and when we consider that a 

 rifle-ball has an initial velocity of 1800 feet a second, it is easy to imagine how inefiicient in 

 comparison with our present weapons must have been those of our ancestors. The wonderful 

 ranges said to have been attained by Robin Hood and William Tell are no doubt mythical. 

 A curious fact concerning the possible amount of the velocity of an arrow is suggested by the 

 formula we have last found, viz. that, supposing their material to be the same, any two bows 

 of similar figure will impart the same maximum velocity, provided that the arrows and cords 

 are also similar and proportional, that is to say, provided that their masses vary as the cube 

 of the linear scale to which the bows are constructed. For, according to the received law of 

 the strength of beams, &c. if F be the maximum strain a bow can support, k the breadth of 

 the bow, and h its thickness at centre, and a its length, 



„ kh' 

 a 

 ••• Fa oc kh' <x the cube of the scale, and m,, m-z, fHa vary also as the cube of the scale ; 



Fa 



m^e + ^ m^ + ^m.^ 



IS constant. 



IT X 



If we had assumed R = F %vn.— v , 



2 I ' 



Y would have equalled 



4FS 



5 



4 



This assumption for the value of JR, which is quite empirical, seems to agree more closely 

 with the results of experiment, than the first assumed value, especially when the cord and 

 arrow are light ; and when in consequence the curve assumed by the bow in motion deviates 

 more from its statical form than when the cord and arrow are heavier. 



If we assume the law of thickness of a bow to be ^ = — ^ , where h is the thickness 



1+1 

 a 



in the middle, or when « = 0, -Z the thickness at any distance s from the middle, y will 



7a^ \T "** 3 ~ JM' ~ 30aV 



to first approximation, and e = '29 nearly. 



But a formula based on a first approximation only will not nearly express the real 

 velocity when a bow is bent to the degree to which it is in practice. 



