y1 = A+B+C+D, y2=A+B.2+C.4+D.8, y3=A+B.3+C.9+D. 27; D2 = C • 2+ D.6, Ya-Y-B+C.3+D.7; D1 = D.6; D3 C.2+D.12; B+C.5+D. 19; 39 = Yo; .. D=¿D ̧, C = (D2 — D3), B=D1−‡D2+‡Ð ̧, and A = hence y1 = yo+(D, − ‡D2+‡D3)≈ +‡(D1⁄2−D3)x2 +‡Ð ̧•x3, (279) or Yz = Yo+x[D ̧ − †D2+‡D3 + x • † (D1⁄2 − D3 + x • ‡D3)], or D2 2 Y2 = Yo + x • — ' + x(x−1) • 2+x(x − 1) (x − 2) • Yx is the function sought. 1.2 For any particular series of numbers, it will be advantageous to calculate beforehand the coefficients of the several powers of x, and to attribute to the terms their proper signs. Thus, for logarithms, the second and third differences, D2, D3, being minus, we have C1 =D1+D2— †D3, C2 = {(D2 — D3), C2 = ¿D ̧ . Let us make an application by requiring the logarithm of 100·345. We have .. log.100'345=y2 = 2 + 345['0043424 — ‘345(‘0000209+ ‘345 '0000001)]. What is the logarithm of 101 7906? Ans. 2007709. Required, the logarithms of 100'1, 1002, 100'3, 100'4, 100'5, 100 6, 100 7, 100 8, 1009. When it is required, as in this example, to interpolate an arithmetical progression of tenths, form (280) may be advantageously modified, as follows. Suppose that we have found the logarithm To adapt this form to the computation of the tenths from 100 to 101, we have * Shortened multiplication. Rule? yr+1=y, +4340·309 — r[4·18 + (r+1) • ‘003]. Yr + 1 = y; 10 Making r = ... log.100‘2 = 2′0008676 ;|log.100‘3=2′0013008;|1005) 20021660. Adapt (281) to interpolate between 101 and 102, and compute the logarithms of 101'1, 1012, ..., 101'9. For returning from the logarithm to its number, we have (280) where, it is to be observed, that C, is a very near trial divisor, though too great. We may therefore find an approximate value of x by dividing by C1, and then, having perfected the divisor, repeat the operation. Given the logarithm 20014956 to find the corresponding num ber. CHARACTERISTIC. 157 If we had divided simply by D1 = 43214, we should have found $346, which is near the truth, and the approximation will be still nearer as we ascend above 100; so that ordinarily it will be sufficient to diminish the given logarithm by the tabular logarithm next below it, and divide this difference by the difference of the tabular logarithms above and below. And when greater accuracy is required, the third difference may generally be neglected, whereby (280) and (282) will be reduced to Further it will be sufficient to correct the divisor by the first digit of the quotient, or the nearest to it, which may be found by inspection. The operation above becomes The Characteristic, or integral part of the logarithm, is not usually inserted in the tables, since it can readily be determined from the relation Hence the characteristic is always indicated by the dis- (284) tance of the first significant figure from the place of units; + if to the left, if to the right. Thus the characteristic, or integral part of the logarithm answer ing to the number 365 is 2; because 3 is two places distant from the units. Therefore, bearing (267) in mind, we have log. 365256229, [See tables]. log. 36'5 = log. 365 = log. 365-log. 10=2656229—1, =1656229, log. 3'65 = log. 165 = (2—2)56229 = 0‘56229, + log. 365 = log. 1 = (2−3 )56229 = 1′56229, log. '0365 = log. 1365 (2-4) 56229 = 256229; &c., &c. 10000= It will be seen from these examples that the logarithm of a number in part or wholly decimal, is to be found in the same way as if it were integral, except the characteristic which is determined by the rule above and may be either plus or minus, while the tabular part of the logarithm is always plus. Note. An operation should be so conducted as to restrict the |