322 On guiding Balloorti 



applied, which can be moved to either extremity of the diagonal 

 BD. 



Having completed tliis apparatus, let the p}ane be placed so 

 that the diagonal AC shall form with the vertical line AF an 

 angle FAC, equal to 70°. Before I relate the experiments, I 

 will attempt to asceitain the direction in which such a balloorif 

 might be expected to mofve. assuming tiie proportions used in 

 the first experiment, when the ratio of the surface of the attached 

 plane to that of a circular plane of the same diameter as the 

 balloon was 1*4 : 1. 



The first object is to find the rclalsive resistances of the baf--' 

 loon and plane, which may be effected thus : Supposing the re- 

 sistance of the balloon to be I, the resistance of a circular plane 

 of the same diameter, and moving ii> a direction at right angles- 

 to its surface, would be 2. Again : The square plane attached t6 

 the balloon opposing L4 times as much surface as such a circular 

 plane, would have a resistance equal to 2*8, if it werfe to lie irf 

 a horizontal position. Bat the breadth of the column of air 

 tvhich it meets is evidently as the sine of the angle of incidence, 

 as is also the force with which it will strike that column: hence 

 the resistance, being in the compound ratio of the surface arid 

 the force, will be as the square of the sine of the angle of inci- 

 dence. In the present case the angle of incidence being 7f)% 

 the resistance of the plane is expressed by 2'8 x sin (70^)*=i 

 2*472, and the resistance of the balloon and plane anited is 

 equal to 1 +2'472 = 3-472. Now it is evident that the velocity 

 of ascent or descent will be a<;celerated until the resistance be- 

 comes equal to the power or weight of the balloon, which may" 

 be therefore expressed by 3*472 also. This being ascertained, 

 let the straight line AB (fig. 2 and 3) be drawn equal to 3*472, 

 which may represent the ascending power (fig. 2) or the' weight 

 (fig. 3) of the balloon : draw another indefinite straight line 

 AC, making the angle BAC = 70'', which will mark the position 

 of the plane. Then, since the angle of reflection i» always equal 

 to the angle of incidence, and the angle, at which tlie plane 

 strikes the air is 70", the angle of reflection would also be 70°. 

 Therefore a straight line AY), equal to 2*472, and making the 

 angle DAC equal to 70°, v/ill represent the resistance of the air 

 to the plane. The balloon and plane being acted upon by the 

 two forces AB and AD, will consequently describe the diagonal 

 AE of the parallelogram ABED, varying from a perpendicular 

 direction by the angle BAE, which is fovmd by trigonometry to 

 be 45^ 12'. Fro-n the figures it is also evident, that the balloon 

 when descending would move in a direction (from A to E, fig.2) 

 directly contrary to that (from A to E, fig. 3) in which it would 

 Inove when eiscending. Tlie preceding rough calculation c«(nr. 



HOt 



