282 Sir William Rowan Hamilton on Fluctuating Functions. 



greater than 27r, and the value oi y being included between those limits. For 

 example, we may assume 



2a =: — TT, 26 = it, 



and then we shall have, by writing a, or, and/, instead of /3, y, and 0, 



1 f» " 



f' — -^ 2(„) _ » J rfa COS (na — nx)f^, (t") 



in which a; > — w, or < tt. It is permitted to assume the function/ such as to 

 vanish when a < 0, > — tt ; and then the formula (t") resolves itself into the 

 two following, which (with a slightly different notation) occur often in the 

 writings of PoissoN, as does also the formula (t") : 



2" \ daf^ + 2(„r, \ da cos {na — nx)f, = -nf^ ; (u") 



h ^ «?«/a + 2(„r. J' da cos {na + nx)f^ =z ; (v") 



2 



'0 



^ being here supposed > 0, but < tt ; and the function/ being arbitrary, but 



finite, and varying gradually, from a = to a = tt, or at least not receiving any 



sudden change of value for any value x of the variable a, to which the formula 



(u") is to be applied. It is evident that the limits of integration in (t") may be 



made to become z^il, I being any finite quantity, by merely multiplying na — nx 



■n . 11. 



under the sign cos., by y, and changing the external factor k~ to ^r^- ; and it is 



under this latter form that the theorem (t") is usually presented by Poisson : 

 who has also remarked, that the difference of the two series (u") and (v") con- 

 ducts to the expression first assigned by Lagrange, for developing an arbitrary 

 function between finite limits, in a series of sines of multiples of the variable on 

 which it depends. 



[12.] In general, in the formula (m"), from which the theorem (c) was 

 derived, in order that x may be susceptible of receiving all values > a and < b 

 (or at least all for which the function /^^ receives no sudden change of value), it 

 is necessary, by the remark made at the beginning of the last article, that the 

 equation 



