Lastly, there is no basis for the behef that the single-heading flight 

 track is always the track of shortest flying time. 



The one contributing factor to the shortening of flying time, which 

 single-heading flight neglects, may be referred to as tailwind shear. Let 

 us assume that the winds are tail winds for a particular flight, but the 

 winds increase in strength to the right of the general path of the air- 

 plane. Then, while the single-heading track would take the airplane 

 along the straight-line from point of departure to destination, yet it is 

 clear that a time-saving would be realized if the airplane were to be 

 flown over a seemingly devious route, to take advantage of stronger 

 tailwinds. 



The above consideration leads us to a study of the minimal or shortest- 

 time flight path. This was the subject of several papers that have 

 appeared in the German mathematical literature back in 193L Several 

 authors, in particular Levi-Civita, obtained the formula for a flat earth 



dd/dt = 8v/8n 

 where dd/dt is the rate of change of the heading of the airplane, and 

 Sv/Sn is the rate of increase of the tailwind across the path of the air- 

 plane. Two or three years ago the United States Air Force Weather 

 Service tackled the problem of extending this theory to the spherical 

 earth. The result of the mathematical investigation was the revised 

 formula for the minimal flight path 



dd/dt = Sv/Sn -\- c -^ V Ss/8n 

 s 

 where the additional term introduces c, the airspeed of the airplane, 

 and s, the scale factor of the chart that is used for representing the 

 earth's surface. If all the headings, 6, are referred to the great circle,, 

 or the shortest path between the airports, then the additional term on 

 the right would vanish. 



In order to begin our considerations of the optimum flight paths 

 over the Pacific, let us choose some hypothetical routes. Let us assume 

 that regular flights are made between the airports San Francisco, Tokyo, 

 Singapore, Auckland, Honolulu, Bogota, and New Year's Island. While 

 these supposed flights are longer than the usual flights of to-da}^ we 

 shall project them into the future when airplanes will be bigger, faster,, 

 and longer-ranged. Let us assume that the flights take place at 20,000 ft., 

 airspeed 300 knots. 



The shortest-time flight paths would be great circles except for the 

 winds. Between Tokyo and San Francisco the winds are westerlies, on 

 the average. Let the airplane be flown on a single heading with respect 

 to the great circle. How should the single heading be found and how 

 should the expected track be plotted ? Briefly, it can be explained as 

 follows : Let us accept an approximate path, such as the great circle 

 itself; Divide this path into legs of about 200 to 400 nm. each. Over 

 each leg there will be an average wind. It will be an easy matter, with 

 a suitable computer, to obtain the •cross-wind component. It is this 

 component which will drift the airplane to the right or to the left of 

 the direct path, the amount of drift being directly proportional to the 

 length of the leg and inversely proportional to the forward speed af 

 the airplane. Let us compute this drift for each leg of the flight and 

 add several computations to obtain the overall drift. This overall 

 drift win then provide the navigator with the proper heading. 



38 



