360 . REPORT— 1880. 



d : (1 -h d), and in any case the problem is simply the analytical expression 

 of any such function of e in terms of the selected function of E. The 

 assumption, that Uq and Uq should coincide, is not a necessary one. 

 The only effect of their not coinciding is to substitute the equation 

 E" = e""'+'' for the simpler form E" = e'"". 



An exposition of the application of this method to symmetrical 

 differences appears to have been first given by Henry Briggs in his Arith- 

 metica Logarithmica, published in 1824. But it appears from his preface 

 that the tables of sines, afterwards published by Gellibrand in the Tri- 

 gonometria Britannica, had been calculated by Briggs, by a more or less 

 complete application of these rules, twenty years earlier.* His exposition 

 is, however, not very well suited to modern use, being rather too much 

 specialised, with a view of suiting his own work. A more general 

 exposition of the method is given by Roger Cotes in his Canonotechnia, 

 sive constructio tabularum per cliff erentias.] This, besides general rules, 

 contains the tabulated coefHcients for the bisection, trisection, and quin- 

 qnisection of the interval. It does not appear that the subject was 

 resumed until recently, when Mr. Woolhouse gave both tables and 

 formulee for the division of the interval by 5 and by lO.J 



According to Lacroix, the method of tabular interpolation for-ordinary 

 differences was first published by Mouton, to whom it was given by his 

 friend Regnaud, in 1670, and afterwards reduced to a general formula by 



Lagrange and Prony. The method is as follows. Let Uq Ui ti.^ 



be a series of numbers, of which the fourth difference may be neglected, 

 and suppose that it is desired to obtain the differences for interpolating 

 two numbers between each. Let the required differences be h, c, d, the 

 original differences being i-epresented in the usual way by Ajt^, A2Mo> &c. . 

 Then, if the interpolated series Uq, iiq + ^j "o + 2 Z< + c, &c. be formed, and 

 the terms representing Uq Ui &c. be picked out, they give 



Ui = Uq + 3& +3g + d 

 U2 = Uo + 6b + 15c + 20c? 

 Us — Uq + 9& + 36c + 84d 

 and differencing these, 



Amq = 36 + 3c + cZ 



A^Mo = 9c + IScZ 



solving these as a est of simultaneous equations, 



c=^AX-|a3«o 



^ = 27 ^'"o § 



* Br5ggs*s\v0rds are 'Kara DiflEerentife, qu8s ante annos viginti mihi maximo usui 

 fuerunt in novo canone sinuiim condendo, in horum Logarithmorum calculo siuit 

 mihi multo melius perspectse et cognitfe.' An explanation and proof of Briggs's 

 method of quinquisection are given by Legendre in the Connaissancc des Temps for 

 1817, p. 219. 



t Published in the same volume with the Harminiia Mensuraritm, by E, Smith, 

 Cambridge, 1722. 

 . X Vidie op. cit. part ii. Also Assitranre Magazirie, vol. xi. p. CI et seq. 



% For the general formula see Lacroix, Traitc de Calcul. vol. iii. p. 43. 



