Sir William Rowan Hamilton on Fluctuating Functions. 307 



1* 3" 5" 177" 179" 

 the cotangents of the ninety angles j-j j-> ^' • • • "X"' ^P' ^^ S'^^n by the or- 

 dinary tables, we obtain nearly, as the average of these ninety logarithms, the 

 number 0,5048 ; of which the double, being multiplied by the Napierian logarithm 

 of ten, gives, nearly, the number 2,325, as an approximate value of the constant 

 ■57. But a much more accurate value may be obtained with little more trouble, 

 by computing separately the doubles of the part (r''^^), and of the sum of (s"^) 

 taken from m= I to m = (x^; for thus we obtain the expression 



# 



■a- = 12 log 3 — 8 log 4 



in which each sum relative to in can be obtained from known results, and the 

 sum relative to k converges tolerably fast ; so that the second line of the expres- 

 sion (x''") is thus found to be nearly = 0,239495, while the first line is nearly 

 := 2,092992 ; and the whole value of the expression (x''") is nearly 



w = 2,332487. (y''") 



There is even an advantage in summing the double of the expression (s*^-^^ only 

 from m =: 2 to m := CO , because the series relative to k converges then more 



OS 



rapidly ; and having thus found 2 \ dati^ar\ it is only necessary to add thereto 

 the expression 



2C (/aN, a-' =12 log 3 -20 log 5 + 28 log 7 — 16 log 8. (z'") 



The form of the function p and the value of the constant sr being determined as 

 in the present article, it is permitted to substitute them in the general equations 

 of this paper ; and thus to deduce new transformations for portions of arbitrary 

 functions, which might have been employed instead of those given by Fourier 

 and PoissoN, if the discontinuous function p, which receives alternately the 

 values 1, 0, and — 1, had been considered simpler in its properties than the tri- 

 gonometrical function cosine. 



[24.] Indeed, when the conditions (t''^) are satisfied, the function p^ can be 



2r2 



