VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 24Q 



The 4th column was thus computed : Let s be the sine of the angle to radius 

 unity, r the resistance at that angle, and suppose r to vary as s m ; then l m : s m :: 



0.2321 : r, hence, s m = J , and consequently m = - g * ; ~~ ° s ' ' -; and, by 



substituting for r and s their several corresponding values, we get the respective 

 values of m, which are the numbers in the 4th column. Now the theory supposes 

 the resistance to vary as the cube of the sine ; whereas, the resistance decreases 

 from an angle of Q0°, in a less ratio than that, but not as any constant power of the 

 sine, nor as any function of the sine and cosine, that I have yet discovered. Hence, 

 the actual resistance is always greater than that which is deduced from theory, 

 assuming the perpendicular resistance to be the same; the reason of which, in part 

 at least, is, that in our theory we neglect the whole of that part of the force which, 

 after resolution, acts parallel to the plane ; whereas, from the experiments which 

 will be afterwards mentioned, it appears that part of that force acts on the plane ; 

 also, the resistance of the fluid which escapes from the plane, into the surrounding 

 fluid, may probably tend to increase the actual resistance above that which the 

 theory gives, in which that consideration does not enter; but, as this latter circum- 

 stance affects the resistance at all angles, and we do not know the quantity of effect 

 which it produces, we cannot say how it may affect the ratio of the resistances at 

 different angles. 



In theory, the resistance perpendicular to the planes is supposed to be equal to 

 the weight of a column of fluid, whose base = 3.73 in. and altitude === the space 

 through which a body must fall to acquire the velocity of 0.66 feet; now that space 

 is 0.08124 in. consequently the weight of the column = 0.1598 Troy oz. ; but the 

 actual resistance was found to be = 0.2321 oz. Hence, the actual resistance of 

 the planes: the resistance in our theory :: 0.2321 : O.1598, which is nearly as 

 3:2. 



I am aware that experiments have been made on the resistances of bodies moving 

 in water, which have agreed with our theory. An extensive set was instituted by 

 D'Alembert, Condorcet, and Bossut, the result of which very nearly coincided with 

 theory, so far as regards the absolute quantity of the perpendicular resistance. Their 

 experiments were made on floating bodies, drawn upon the fluid by a force acting 

 on them in a direction parallel to the surface of the fluid. There can be no doubt 

 but that these experiments were very accurately made. The experiments here re- 

 lated were also repeated so often, and with so much care, and the results always 

 agreed so nearly, that there can be no doubt but that they give the actual resistance 

 to a very considerable degree of accuracy. In our experiments, the planes were im- 

 mersed at some depth, in the fluid ; in the other case, the bodies floated on the 

 surface ; and I can see no way of accounting for the difference of the resistances, 

 but by supposing that, at the surface of the fluid, the fluid from the end of the 

 body may escape more easily than when the body is immersed below the surface ; 

 but this I confess appears by no means a satisfactory solution of the difficulty. The 



VOL. XVIII. K K 



