116 Proceedings of the Royal Irish Academy, 



number is due to the fact that (66) has to be multiplied, instead of divided 

 as is the case with (23), by 



(-^)' + /i3«^)i(-/-//3a^)l, 



in order that it may represent, for large values of p\ the true period-equation. 



Accordingly, when la is very small, the period-equation has one root not 

 given hy (15). This root gives a value to i^ which is itself very small and 

 diminishes indefinitely with la. In fact, if la is zero, one value of p' is zero ; 

 this may be seen by noting that when la is zero, Yi = Yz, in the notation of 

 (39) ; ij' will now be zero if Pi = Fj = ; and it is evident that these values 

 satisfy the period-equation, after its division by Fj - Y^, or an equivalent 

 differentiation, which is a necessary preliminary. If, returning to (1), in it 

 we replace / by zero, we do indeed obtain a root, p' = zero, corresponding to a 

 disturbance in which S is constant, in time and in space. 



Thus, if la be small enough, here again all the disturbances are aperiodic, 

 and all the roots are accounted for by (15), with the exception of this one, 

 which we may regard as also included in (15) on making r zero. 



It is readily seen that a value oi p' occurs at G (fig. 2, p. 108), whenever at 



this point 



/_|(«) = 0, i.e. u = {r-K + 57r/12)*, 



or /§ {u) = 0, i.e. u = {m + 137r/12)*, 



r being zero or any positive integer. The former set are double roots ; and it 

 may be proved much as in Art. 19 that these are the only double roots. 



We may trace, as in Art. 20, the effect of diminishing the wave-length in 

 the direction of flow on the nature of the roots. When la is exceedingly 

 small, one value of p^ is close to (fig. 2), and all the others to the left of C ; 

 as I is gradually increased, all the roots move towards C until the expression 

 (51) becomes equal to the lowest zero of J-^{x), at this stage two values of p' 

 coincide at C. On increasing I still further, these two roots become complex, 

 and there is now no value between C and until (51) becomes equal to the 

 lowest zero of J^(x) when a value of ^' passes C, to return to it and in coinci- 

 dence with another become a double root when (51) becomes equal to the next 

 zero of J-^(x); after this these two become complex and different; and so on. 



The greatest wave-length in the direction of flow for which a disturbance 

 can be oscillatory is thus 2,Tr/l, where 



{32/3/ftV(27v/3i.))2 = the lowest zero of /-|(^jj) = 1-2. (67) 



There are a finite number of complex roots, those whose imaginary parts 

 are positive being given, when not too near 0, by the approximate equation 



e^'i - 'ic""! = 0, 

 or, tti = rjT + 7r/4, (68) 



