Omi — Stability or Instability of Motions of a Viscous Liquid. 105 



complex roots for which '£> lies near one or other of the productions 

 mentioned.) 



We have then to trace the change of argument of e"2-"i - g"i-«2 as 'p' 

 describes this circular contour. It will be more convenient to suppose jf 

 to start from, and stop at, the poinb of the circle midway between AL, A'H . 

 From (27a), (28) it is seen that, as j) describes the contour, the real part 

 of % - U\ starts from an initial value zero, is continually positive, and ends 

 with the value zero, while the imaginary part continually increases from 



- (2r + 1) 7r/2 to + (2r + 1) 7r/2. 



Thus, of the vectors e"2""i, e"i""2, the former is throughout the greater, 

 except that their initial values are equal ;* the former revolves in the positive 

 direction, and the latter in the negative direction, each through an angle 

 (2r + 1) TT ; owing to the former being throughout the greater, the vector 

 g"3-wi - gWi-«2^ which is their difference, follows the direction of the former, 

 oscillating about it, but never rotating round it,t making, indeed, always 

 an acute angle with it. As the initial direction of this difference is the 

 same as that of e^a-^i^ and as the same is true of the final directions, the 

 total angle through which the vector difference rotates is the same as 

 that through which e"2-«i rotates, i.e. a positive angle (2r + 1) tt. Thus, 

 while y describes the circle, the argument of the left-hand member of (29) 

 increases by [Ir + 1) tt. But the points A, A' are zeroes of the left-hand 

 member of (29), extraneous to the proper period-equation; the increase in 

 the argument of the extra factor (uiU^)^, or in {- ij' ^- l^ai)ki- p - l^ai)\, is tt. 

 Subtracting this we obtain an increase of 2r7r as that depending on the number 

 of zeroes we wish to find ; hence their number is r. But all the zeroes have been 

 proved to lie between the lines AL, A'L\ By giving v the values 0, 1, 2, etc., 

 in succession, we see that there is no zero to the right of the arc of the first 

 circle r = 0, and that there is one and only one zero in each of the quadrilateral 

 spaces bounded by two consecutive circles and the parallel lines. And it 

 has been already shown that in each such space there is one real zero given 

 approximately by ii^-Ui-=-rTti\ hence, under the circumstances referred to at 

 the beginning of the Art., this approximate equation gives all the zeroes. 



And the same argument shows that whatever the value of la, if r is large 

 enough, the number of zeroes lying inside the circle referred to in (28) is i\ 



"''' But opposite, and the same statements hold, of course, for their final values. 



t It is important to note that in the first ami last quadrants of the circular contour the real part 

 of M2 - Ml changes more rapidly (and in the first and last portions exceedingly more rapidly) than the 

 imaginary part, so that when the vectors, which are represented only approximately by e'V'i and 

 e"r"2, are in the same direction, even for the first and the last times, the former is very much 

 the greater. 



