246 SYMPOSIUM ON AERONAUTICS. 



We have, for example, for a head gust it^, 



.i28X^+i-i6X'^ + 3- 385X + -9i7 

 ^ (X - /x)(X + 4.18 ± 2.43i.)(X + .0654 ± .iS7i) "^' ^-^^ 



where the double sign stands for two factors, and u^ = J{i — e~*), 

 to take a particular case. The residues at each point are merely 

 the values of the fraction when one of the factors, the one which 

 vanishes at that point, is thrown out of the denominator. In the 

 first case for i = c*" we have as residue of ^e^* = $: 

 at \^fx = o, 



-9 17 



(+ 4-i8 ± 2.43i)(.o654 ± .i87i) 



at A = — 4.18 — 2.431, 



.i28X=' + i.i6X2 + 3.385X + .9i7 



(-4.i8 + 2.437")(-4.86-/)(- 4.12 - 2.437: ± .1877') ' 



at A = — 4.18 -|- 2.43J, the conjugate imaginary expression. And 

 so on. To treat e~^ we should have : 

 at A = ^ = — I, 



_ - .128 + i-i 6 -3- 38 5 + -9 17 . 

 (3.18 ± 2.437') (.9346 db .i87i) ' 

 and so on. 



As the calculation with imaginaries involving squares, cubes, 

 products, and quotients is by no means simple, it is clear that to get 

 the solution for u will be reasonably hard work — much harder than 

 to find the particular solutions which for the simple gust involved 

 only real numbers. It may be admitted that to work any one gust 

 the labor will probably be much less than by my method of determin- 

 ing formulas for the constants of integration in terms of the initial 

 values of the particular integrals. But as far as I can see, Brom- 

 wich's method is of no particular advantage if we desire to calculate 

 the effects of a large number of gusts /(i — C'' ) of various degrees 

 of sharpness both head-on, up, and rotary. When we came to cal- 

 culate a periodic gust we found that we were involved in powers and 

 products and quotients of complex numbers, and it is probable that 

 the work we did in finding the particular integrals was comparable 

 with that required for the present analysis. 



