WILSON— AEROPLANE ENCOUNTERLNG GUSTS. 221 



We note the very different values of p thus found, namely, o, 0.15, 

 0.776, according to the choice of case. If in Case i, we had taken 

 /> = .I5, the amplitude ratio would have been about .7 instead of i ; 

 if p = .y'j6 had been assumed, the ratio would have been .37 instead 

 of I. 



Case 5. — If we desired to maximize z we should have had to 

 treat 



_ i 3-95^+-598//> ^.^^ 

 i .598 — p' -\- 4..og8pi 



which would have given an infinite amplitude ratio for p^o. 



17. Now if we turn to the free machine and try to maximize 

 f6dt instead of $, we have to maximize 



.04263- + -OiG'p'" 



(.039242 - p^y + .i3o82/)2 



instead of (5, §6). The value of p^ is about .0307 and of p about 

 .175 instead of .2 as before. The amplitude ratio is then only 

 slightly in excess (about 4 per cent.) of that previously found — in 

 other words the numerical values are such that resonance for 6 and 

 for fOdt, which is the preponderating term in the expression for s, 

 occurs for considerably different values of p, but the effect is about 

 the same. This may be regarded as validating our procedure (§6) 

 in maximizing 6 instead of fOdt. 



18. To sum up this discussion of resonance as applied to the 

 aeroplane we may say that the frequencies which produce " reso- 

 nance " depend largely upon the quantity in which the effect of reso- 

 nance is to be sought and that the frequency which makes for a 

 strong resonant effect in one quantity may make on another an effect 

 much weaker than the maximum — or it may not. 



19. There remains to discuss the question whether the eft'ect of 

 resonance is practically serious, i. e., whether as in the case of the 

 motion of the machine, above treated, the effect fails to make itself 

 felt until after so long a time that the pilot would be entirely able to 

 deal with it or the wind would really have in all probability ceased 

 to be periodic with the period required. 



Now in order to insure that resonance is effective both of itself 



