234 SYMPOSIUM ON AERONAUTICS. 



•894Q3 + 3-92C33 -h 85.74C34 = o, 



•894C14 — 85.74^33 + 3-92C3, = o, 

 and 



Ci3 = i04iC3 + s64.8C4, C"33 = — 6.37iC,3 + 10.56C,, 



Ci, = — 564.8C3 + 1041C4, C3, = —io.56C3 — 6.371 C04. 



The solutions therefore, so far as concerns the complementary 

 function, are 



-j- C04 sin 1.2150, 

 v = — 8.^26Co^e-'>-^^''^ + 3-797Co^-'-''-* 



+ ^-•^-*^^*[(io4iC3 4-564.8C.J cos 1. 215; 



4- (— 564.8C3+ 1041C0J sin 1.215^], 

 y==o.257iC2i^-"-^^" -\- o.oosSgyC.^e-^-^-^-* 



+ ^"'''''[(— 6.37i<^23 + iO-56CJ cos 1.215? 



— (10.56C03+ 6.371) sin 1.215^]. 



34. These equations determine the relative magnitudes of the 

 various sorts of natural motion. 



The first term is the slowly amplifying divergence, this machine 

 being slightly unstable laterally. If a side gust is such as to induce 

 a lateral velocity of — 8.326C21, it induces a bank of C21, an eighth 

 as much in radians or seven times as much in degrees. It is 

 therefore clear that only very small values of Co^ are admissible for 

 safety. The second term, corresponding to the rapidly damped 

 motion, shows such rapid damping that it can hardly be of impor- 

 tance, except for possible strains on the mechanism, unless Coo is 

 so large that the whole work is inapplicable because of the failure 

 of the motions to be small. 



The trigonometric terms show that the oscillation in v will be of 

 great amplitude compared with that in <f), the factor being about 

 1200 when cfi is in radians or 20 when (p is in degrees ; even the 

 oscillation in r will be over 12 times as great as in (f>. In other 

 words, the machine may have a large oscillatory side-slip or angular 



