20 Rayleigii, TJic MecJimiical Principles of FligJit. 



When frictional forces are included we may use equa- 

 tion (i 8), merely substituting Fsina for U. The problem 

 already considered of making U a minimum is still 

 pertinent, since t/ denotes the rate of vertical descent. By 



(19), (20), (21) 



sin°"a = ^^, sin- « = -„:,= -^ • • • (25), 

 KO V ' 3(>.-o 



so that, i) and n being small, 



o = fe, 9-a = ia=l-0 . . . (26). 



This result, due to Penaud, shews that when the rate of 



vertical descent is slowest, or when the time of falling a 



given height is greatest, the slope of the plane to the 



horizon is downwards in front, and equal to one-quarter of 



the slope of the line of motion. The actual minimum rate of 



vertical descent is given by (20). This rate is relative to 



still air. If there be a wind having a vertical component 



of the same amount, the course of the plane may be 



horizontal. 



Another slightly different minimum problem is also 



treated by Penaud, in which it is required to determine 



how Jar it is possible to glide while falling through a 



given vertical height. From (9), (17), (23;, we have in 



general 



sin -a 4 nUS , > 



sinO= -■ — (27), 



SU1« 



When y is a minimum by variation of a, 



sina = 4sine= ^O^/k-^) (28). 



In this case the plane bisects the angle between the hori- 

 zontal and the direction of motion. 



In the flying machines of Penaud, Langley, and 

 Maxim, the propelling force is obtained by a screw, 

 acting like the screw-propeller of a ship. A rough theory 



