17(3 New Tables of the PruhahUitij Inlajral 



We need only use inside the square brackets the numerical valucs of Ö and of 

 /ii)«„, h-iy-Ut,, ..., if we put tlie sigu of 6 JiDu^ outside, and follow the rule of 

 signs inside. The rule is that a minus sign niust be iiiserted, if 6 be positive, when- 



ever there is a change of sign in the scries hDu^, h^D'u^ or, if ö be negative, 



whenever there is no change of sign in this series. 



Thus, in Table I., taking u to be ^ (1 + a), we havc for a-o="40 (omitting 

 decimal point) 



3G826i -147 -3 

 hüll, h-D-u, h'I?Uo 



3G827 -147 -3 



and thcrefore, for a;="40+01ö, 



i (1+ a) = 10-- (6554217 + 6 [36827 - \d (147 + 'Ö . 3)]). 



For a;=-4ü--01öwe should replace +6, -W, +16, hy -0, +\e, -^0. The 

 formulffi giveu by (3) and (4) would be, for a; = '40 + 'Olö, 



Hl + «) = 10- (6554217 + 36753(? + 150 ^^^ - 3 ^^i:^^^?^^) , 

 Hl + a)= 10-' (6554217 + 36753Ö + 150 ^JlZL^ + 147 t(k:J^)\ . 



There is no difllculty about tlie divisions represented by the coefficients 

 i, ^, ... of in (6); but, if wc wish to avoid thcm, we may calculate lv'D^u^j2\, 

 h^Ifiu^/3 !, ... and write the forniula 



u = /*o + Ö [hDu, + (H=^=«o + QJi'LPuc, + ...)!] (7). 



3. Inverse Interpolation. In most cases N^ and Nn [= iiV"(l + «) and ^N{\ - a)] 

 are known, and we requirc x. If Hl + ") ''^^'•^ Hl ~ '*) ^^^ both less than 90, we 

 can use Table III., which givcs x in terms of a = {Ni — N^jN. But, if either 

 ^(1 + a) or Hl~*) l'ß greater than 'OO, we niust use Table L, by inverse 

 Interpolation. By (6) we have, if a; = a;, + dh, 



e = {u- »„) - [hDu, + hß {IrD'u, + W {h'D'Uo + ...))] (8). 



or, if x = Xo— 0h, 



= (u„-ii)^[hDu„-^d [k''I)^Uo-^0(hWit„-...yf\ (8a). 



The value of 6, and thence that of x, is obtained by successive appro.ximations. 



Suppose, for instance, that 



Hl + a) = -654. 



If .r = •40-01 ö, we have from Table I. 



ö= 14217 H- [36827 + iö (147 - 0\]. 



