Orr — Stability or Instahility of Motions of a Viscous Liquid. 73 



Art. 23, p. 113, contains two fundamental equations showing how to 

 discover the coefficients of the quasi-Y owxiev expansion of an arbitrary 

 function of y in a series consisting of the infinity of F's which correspond to 

 given vahies of I, 7i ; it seems reasonable to assume the possibility of such an 

 expansion ; I am quite unable to prove it. I have failed in the endeavour to 

 apply this analysis quantitatively to the case of a disturbance of simple type, 

 as was done in Part I., Chap. I., Arts. 4-8. 



In Art. 24, p. 115, a brief reference is made to the case in which the 

 boundary-conditions V^v = are replaced by djdy . V^v = 0. 



The much more difficult case in which the boundary-conditions are 



-y = 0, dvjdy = 

 is taken up in Art. 25, p. 117 ; it is proved that the imaginary part 

 of p lies between the same limits as before. I have failed, however, to 

 obtain any direct probf from the differential equation itself that 23 has a 

 negative real part, and also to obtain any equations by the aid of which 

 the Fourier analysis of an arbitrary disturbance can be performed. There 

 is frequently a connexion between these two questions ; a fundamental 

 equation of Bessel-Fourier analysis,* for instance, serves equally to prove 

 that all zeroes of the Bessel function of order greater than - 1 are real ; 

 and, though equation (63) of Art. 23 does not show the roots to have a real 

 negative part with the boundary-conditions V^-y = 0, the two results have 

 been obtained by similar methods. Probably some simple proof that i3 has 

 a negative real part in the present case will be discovered ; but it seems 

 possible that no simple theorem relating to Fourier expansion may hold. 

 Similar difficulties may arise to a certain extent, even for a system having 

 only a finite number of coordinates ; in some such cases the proof of stability 

 for fundamental disturbances is much more difficult than that of the reality 

 of the roots of the determinantal equation which is met in the corresponding 

 problem of displacement from equilibrium, and the period equation may 

 have to be examined as carefully as any other algebraic equation, the fact 

 that it arises in a dynamical problem being regarded as a mere accident; 

 also, when, in steady motion, the fundamental determinant is unsymmetrical, 

 and there exist forces of resistance proportional to the velocities, no rule 

 appears to be known for abbreviating the labour of solving the simultaneous 

 simple equations which determine the coefficients of the fundamental 

 disturbances making up a given initial one. 



* I. e. the equal ion 



Jn{K)-) Jn{\>-) rdr = 0, 

 Jo 



where k«, \a aie different zerous of Jn(x), and « + 1 is positive. 



