Oer — Stahilitif or Instahiliiy of Motions of a Viscous Liquid. 87 



coordinates is infinite would probably be a problem of considerable difficulty. 

 Consider a system possessing only one coordinate, and governed by the equation 



(Fx/df- + (a + h) dx/cU + abx = X, (21) 



where, when t is negative, X is zero, and, when t is positive, X = e'''\ c being 

 positive, or having its real part positive. The solution in which Sit t = 

 X and dx/dt are zero, is known to be, for positive values of t, 



(a - h) (b - c)(c - a) x = {h ~ a) e'"* + (c - h) e""^ + (a - c)e-^K (22) 

 By means of the equation 



f{t) = TT"^ dw f{u) cos w {u - 1) du, (23) 



J J -a> 



Fourier analysis of the disturbing force gives 



c cos (ot + (i) sin cot 



The solution of 



G' + m" 



doy. (24) 



d'xjdf + {a + h) d.x\dt -{ abx = c cos wt + oj sin wt, (25) 



which is of the same period as the disturbing force, being 

 (^ _ ^\^ ^ ^^^'^ ~ ^') °°^ ^^ + (^^ + c) w sin o)t (be - w") cos w(( + (& + c) w sin (ot 



(26) 

 the integral solution obtained in the way indicated is accordingly 



(b - a) TTX 



(ac - h/) cos <jjt + {a + c) w sin o)t 

 (a^ + w^) [& + i^') 



{be - w^) cos (Dt + (b + c) (i) sin it)t 



' {b^ + o,'){e + u>') ' 



or 



c cos u)t + M sin (i)t 



(a -b)(b~ c) (c - a)Trx = (b - a) 



dis 



h O.Ct!^ i.tf. 4- /.I Rin t.if. 



duj. 







, 7, f" « cos wi^ + w sin &)(( , , X 1 " S COS w^ + w sin wt 

 + {c-b)\ — dio+(a-c)\ r-. . 



(27) 

 The first integral on the right is zero when t is negative, and ire'"^ when t is 

 positive ; if the real part of a is positive, the second integral is zero when t is 

 negative, and 7re~"* when t is positive ; but, on the other hand, if the real part of 

 a is negative, it is zero when t is positive, and tts"** when t is negative ;* while 

 it is infinite if the real part of a is zero ; and similar statements hold for the 

 third term. Thus the value of x as given by (27) agrees with the correct 



* These alaieinuuis are equivalent to equation (2i), a and c beiuj^' intercliaiiged wkere necessary. 



[12-J 



