280 Sir William Rowan Hamilton on Fluctuating Functions. 



^^^=:J^°rfaj'rf/3p,,; (g") 



so that, as before, 



\ 1 w 



TSr — TIT -y- w . 



Finally, when x is outside the limits a and b, the double integral in (b) vanishes ; 

 so that 



b *" 



= f dai fl?/3p^(„_x)/„, if ^ < a, or > 6. (h") 



And the foregoing theorems will still hold good, if the function y^ receive any 

 number of sudden changes of value, between the limits of integration, provided 

 that it remain finite between them ; except that for those very values d of the 

 variable a, for which the finite function y^ receives any such sudden variation, so 

 as to become =y^ for values of a infinitely little greater than a, after having 

 been =y^^ for values infinitely little less than a, we shall have, instead of (b), 

 the formula 



-T + -r = C da f rf/3 P,(„_„,/„. (i") 



[11.] Ifp<.be not only finite for small values of a, but also vary gradually 

 for such values, then, whether a be positive or negative, we shall have 



lim 



and if the equation 



_ .N„a- = P„; . (k") 



a = 



N._. = (1") 



have no real root a, except the root a = a:, between the limits a and b, nor any 



which coincides with either of those limits, then we may change/^ to ^^ -f^, 



in the formula (z'), and we shall have the expression : 



/x = '=r~'Po\ c^aN«(„_x,N„_!^/„. (m") 



Instead of the infinite factor in the index, we may substitute any large number, 

 for example, an uneven integer, and take the limit with respect to it ; we may, 

 therefore, write 



