Sir William Rowan Hamilton on Fluctuating Functions. 291 

 of a greater tlwn 0, but not greater than /. The equations (b^*") give 



(]8cosj8/+»/sin/30\ fl?asinj3a/„= . (c^'') 



(P sin §l—v cos /3/) \ da cos ^a/, - cos ^a (/„ ^ , +/„_,) ; 

 SO that 



{p sin pZ — 1/ cos /)/) \ da cos /)o/„ = cos pa(f,+, +/„_i), (d'O 



if p be a root of the equation - 



p cos pl-\-v sin pi = 0. (e^O 



This latter equation is that which here corresponds to (g) ; and when we change 

 ^ to p-\-y, 7 being very small, we may write, in the first member of (c^''), 



j3cos/3/-l- *'sinpZ = 7 [(1 -\- vl) cospl — pl^m pi}, (f-"') 



and change j3 to /j in all the terms of the second member, except in the fluctua- 

 ting factor cos §a, in which a is to be made extremely large. Also, after making 

 cos /3a := cos pa. cos 701 — sin pocsin 7a, we may suppress cos yac in the second mem- 

 ber of (c^*^), before integrating with respect to 7, because by (d^^) the terms 

 involving cos7« tend to vanish with 7, and because 7"' cos yx changes sign with 



7. On the other hand, the integral of is to be replaced by tt, though 



it be taken only for very small values, negative and positive, of 7, because « is 

 here indefinitely large and positive. Thus in the present question, the formula 



/, = ! . 1™ • C c/psin^.r(''°(/asinpa/„ (g^O 



TT a = CO Jo ♦^i-a 



(which is obtained from (a'") by suppressing the terms which involve cos /3jr, on 

 account of the first condition (b^''),) may be replaced by a sum relative to the real 

 and positive roots of the equation (e^'') ; the term corresponding to any one such 

 root being 



{1 -\- vl) cos pi — plsmpl* ^ ^ 



if we suppose p > 0, and make for abridgment 



2 p 2 



