302 Sir William Rowan Hamilton on Fluctuating Functions. 

 in which 



And if a root v of (a''^) should satisfy the condition 



the corresponding term in the first member of (t) would then be 

 in which 



Finally, if a value of ^ + j/ satisfy the conditions (k''^), and if the function y 

 undergo a sudden change of value for this particular value of the variable on 

 which that function depends, so thatyzr^^^ immediately before, andy=y^ imme- 

 diately after the change, then the corresponding part of the first member of the 

 formula (t) is 



And in the formulas for w,, ts,, w\, it is permitted to write 



N„ + p,a-> = C dt Pta + fi,' (s''0 



[22.] One of the simplest ways of rendering the integral (e^") determinate at 

 its limit, is to suppose that the function p„ is of the periodical form which satisfies 

 the two following equations, 



p being some given positive constant. Multiplying these equations by da, and 

 integrating from a = 0, we find, by (a"), 



N_a + N„ = 0, N„+j, + N„ = N,; (u''0 



therefore 



Np = Np + N_p = 0, (v''0 



and 



N„ + p = — No, N„ + jp = N„, &C. (w''0 



