WEBSTER— DYNAMICAL ASPECTS. 167 



these six motions affects all the others, as already shown for pitch- 

 ing and rising. In treating this problem we use differential equa- 

 tions invented by Euler for problems in which we have to do with 

 rotating axes of coordinates, and we are thus able to find the mutual 

 connection of the different sorts of motion. Now if the disturb- 

 ances are small, we are able to use the method introduced by La- 

 grange in his famous " ]\Iecanique Analytique " for the treatment of 

 small oscillations, which leads to the introduction of an algebraic 

 equation of degree twice as great as the number of degrees of free- 

 dom of the system, in our case six, so that the equation would be 

 of the twelfth degree. On account of symmetry, however, our 

 equation reduces to degree eight, and falls apart into two equations 

 of degree four. It is useless to undertake the general solution of 

 these, but when we have the constants of a given apparatus, as 

 determined by experiment, it is possible to solve the equations arith- 

 metically with any desired degree of approximation. This is what 

 has been done by various investigators, like Bryan and Bairstow 

 in England, and Professor E. B. Wilson here. In fact when this 

 work has proceeded to a certain extent, it is no longer necessary to 

 have recourse to learned mathematicians, but it may be farmed out 



Fig. 7. 



to computers, so as to be greatly expedited, and thus the design of 

 machines may be greatly improved. I may say that machines gen- 

 erally gain more stability with greater speed, and that too great 

 stability is not desirable, as it would lead to difficulty in steering or 

 rising. At any rate the theory has now arrived at such a stage that 

 we may hope to avoid such accidents as formerly occurred in great 

 numbers owing to improper design. 



I will conclude with a simple experiment showing the intrinsic 

 stabihty possessed by a very simple aeroplane such as I learned to 

 make when a schoolboy, which I am able to fold from a piece of 

 paper before your eyes and to throw with a good deal of accuracy. 



