Mr. Tredgold on the Theory of Hydro-dynamics. 13 



to the effective resistance as AE is to AB ; consequently, the 

 actual pressure of a fluid on a plane being the same, its re- 

 sistance will vary as AB, or as the sine of the angle CAB. 



The omission in the demonstration of Prop. 34 of the Prin- 

 ciples consists in taking the resistance of a particle in a direc- 

 tion opposite to the motion of the body as its whole effect. 

 Whereas, in consequence of the continuity of the fluid, re- 

 flection cannot take place, and the actual resistance in a di- 

 rection perpendicular to the surface must be composed of two 

 parts, viz. — the force in the direction of the motion, and the 

 force by which reflection is prevented. This leads us to a valu- 

 able theorem in inquiries of this kind. 



Prop. — If a plane surface moves with an equable velocity 

 in a fluid, the height of a column of that fluid which would 

 generate that velocity is to the height of a column equivalent 

 to the resistance of the fluid as the square of the radius is to 

 twice the square of the sine of the angle the plane makes with 

 the direction of its motion. 



The resistance is as AB, but the force of the fluid is as EA ; 

 and, if this force be decomposed, we have its two parts A6, and 

 Aa, of which Kb is proportional to the height of a column of 

 the fluid which is due to the velocity of the plane's motion ; 

 and Aa the resistance to reflection. 



But the angle of reflection being equal to the angle of inci- 

 dence, and the triangles similar, 



Rad. : sin CAB:: Ab:\KE; 



i k -,-, 2 Ab x sin CAB 



whence AE = : . 



iad. 



Again, rad. : sin CAB : : AE : AB, or AB = 2Afcxs?n2CAB . 

 ° rad. 3 ' 



and, consequently, Ab : AB : : rad 2 : 2 sin 3 CAB. 



When the plane is moved in a direction perpendicular to 

 its surface, Ab becomes half AB, or the resistance is double 

 the weight of the column due to the velocity. This distinc- 

 tion between the resistance and the head due to the velocity is 

 most important ; and the fact of its coinciding with the result 

 previously obtained for the particular case when the motion is 

 perpendicular to the plane, will have its weight in procuring the 

 theory of hydraulics here laid down some degree of attention. 



The comparison of the resistance of a cylinder and a globe 

 requires that a further research should be developed, as the 

 plane's motion and the pressure of the fluid which follows must 

 be taken into consideration. Were we to neglect these, or 

 simply wished to compare the direct resistance of a globe and 

 cylinder of equal diameters, the ratio would be as 2 is to 3, 

 instead of as 1 to 2. 



IV. An 



