716 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxiv 



having all terms alike except the last two in each. Taking n = 2 or 3 

 and making suitable assumptions, we find that these functions have two 

 common linear factors (pp. 148-50, with changed notations). Besides 

 employing roots in three or more arithmetical progressions, leading to a 

 solution of degree 7 (p. 152), various special methods are used. 



Escott, after reading the proof-sheets of this chapter, pointed out its 

 relation to the derivation of formulas for the computation of TT : 



tan- 1 - - 4-tan- 1 - - H ---- = tan- 1 , 



where X is a real polynomial in x whose degree equals the number of frac- 

 tions in the left member. Since 



a 1 y+ai 

 tan~ 1 - = log- -., 

 y 2i yai 



it suffices to have (x+a+ai)(x+l3-\-bi)' -=X+pi. Of the polynomials 

 m, n in the above problem on logarithms, we may employ here those con- 

 taining only odd powers of x and a constant term. If in Delambre's 56 ex- 

 ample we replace a by ai and b by bi, we have 



(x + ai) (x + bi) (x ai bi) = x 3 + (a 2 + ab + 6 2 )# + ab (a -f- 6) i, 



, a . ,b a+b ab(a+b) 



tan- 1 -+ tan ] --tan x - -=tan 1 . , . . . . 

 xx x x 3 +(a z -\-ab+b 2 )x 



By the former we have a product of factors like # 2 +a 2 expressed as a sum 

 of two squares (cf. note 13, p. 382 of Vol. I of this History). Escott noted 

 that his 63 general results include as special cases Goldbach's 1 and Euler's 2 

 formulas, the first identity by Nicholson 8 , the two formulas by J. H. Tay- 

 lor, 11 as well as the following (after reducing each term by such a constant 

 that the sum of the terms in either member becomes zero 16 ) : Gerardin's 13 

 2, -, 47, Gerardin, 17 - 42 Tarry, 17 Miot, 40 and Aubry. 46 



In Sphinx- Oedipe, 10, 1915, 30, occur two examples of two sets of five 

 numbers having equal sums of &th powers for &=1, -, 4, the numbers 

 being functions of six parameters. 



