108 Proceedings of the Royal Irish Academy. 



on multiplying by da, and integrating between the limits and 1, we obtain 



J 

 by supposition the left-hand member is zero, while the integrand on the 

 right, being the product of conjugate complex factors, is essentially positive ; 

 accordingly Yi' and Y^ must be equal; and, on substituting in succession 

 y,, Yi in (47), we evidently return to the special exceptional cases again. 



Aet. 20. The March of the Roots, as the Wave-Length, in Direction of Flow 

 decreases. A finite Number of Disturhances become Oscillatory. 



In fig. 2, let be the origin, A, A' the points ^lai, - (ilai, and 6' the point 



-i3/«/\/3. 



As proved in Art. 19, when a double root occurs, the value of j)' is represented 

 by the point C. 



I desire to make use of some expression for the error in terminating, after 

 an assigned term, the divergent series which occur in connexion with the 

 Bessel functions ; a partial statement as to this error has been made in 



Fig. 2. 



connexion with equation (9) ; it may now be completed by stating that, 

 in that equation, if the argument of a^ is ±(77-7), 7 being acute, one 

 form of the multiplier there alluded to is 



cosec (9 + 7) (sec 6')2+^'+^ 



where 6 is any acute angle such that 6/ + 7 is also acute ; in the case in hand 

 we may conveniently take 9 to be zero, and use the theorem that the error is 

 less than the next term multiplied by cosec 7. And as when h = ^, ^ - k + s 

 is positive, even when s is zero, we may use this form of remainder after any 

 number (even zero) of terms. When p' lies between and 0, the argument 

 of Ui lies between 7r/2 and 37r/4, and that of V2 between - 7r/2 and - 37r/4, so 



