of Fluids, compared with the best Experiments. 253 



17. According to this investigation we have therefore the 

 following order of resistance. 



A cylinder with flat ends 3 or 1*00 



A cylinder with the hind part a hemisphere 2*5 or 0-833 



A cylinder with the fore part a hemisphere 1 *8 or 0*60 



A sphere 1*3 or 0-4.33 



These ratios are likely to be altered a little by the effect of 

 friction ; and if the cylinder be reduced in length till it becomes 

 a thin plate (fig. 9. and 10.), a still greater alteration is caused 

 by the interference of the two motions of the fluid. The ef- 

 fect is easily observed by moving differently-formed bodies in 

 water. 



18. When a cylinder moves in a direction perpendicular to 

 its axis (fig. 7. and 8.), making d its length, we have 2di = 



the fluxion of its area, and — = sin a ■= sin c; hence, 



r 7 



The fluents are, when z is the arc of the curve, 



Idv* / 3r*z —{r-x). (3r*y + 2y») 2 r*-2 (r-x)3-3 y « (r— x) v. 



4g \ 4^ * 3^ / 



= Hx2rfi. 



In this form the equation applies to the curved ends of 

 canal boats having flat bottoms, the curves being usually por- 

 tions of circles ; and an approximate equation in the following 

 form is easily applied. 



Let mr = half the breadth of the boat, r being the radius 

 of curvature; then 



2rfu* f 3 z -r (1 -m) . (3-4m) V 2^ r( 2-2(l-m ) ( ( 1 - m)* + 3^) ? _ 



Ag i 4 + 3 3 



( 2dHi'. In the ordinary boats m ='125, and therefore, 



Ag > 4 / 



From these equations I have been enabled to compare some 

 experiments made by Mr. Bevan on the power required to 

 draw canal boats ; which will be detailed after treating of the 

 friction of bodies moving in fluids. 



When a perfect cylinder is the form of the body, then r -=.x 

 and p = 3*14159 7-, we have, 



3p j 2 \ 1-8448 v* 



4g \ 8 + 3 / 



H. 



Ag 



19. 'Of the Friction of Fluids* — A series of experiments on 



the 



