When h or Ax diminishes indefinitely, it is clear that > increases indefinitely. When, therefore, this takes place, log (1 +) "becomes, from the definition above, log e = 1. At the same time, when Ax or h dwindles indefinitely, Ay Ax be I = as § 41. ILLUSTRATIONS. Some interesting and instructive results may be derived from these equations. On the same base as before, plot the logarithms as found in ordinary tables. A curve will be obtained similar to the other in general character, but flatter. It crosses the line OX at the same point as the other. Its may be approximately found from the slope at this point tables, for we have log10 I = 0'000,0000 log10 1'0001 = 0'000,0434 1 See note (9), p. 194. Ay Ax' which, when Ax is so small as o'0001, will be very It is interesting to notice that, in an ordinary book of logarithms, the ‚height of the derived curve of the curve of ordinary logarithms is given by the side of the tables, so as to enable any one using the tables to "interpolate.' This height is called "difference" in the tables. The principle made use of in the calculation of intermediate logarithms is The value of is given as a difference." бу dy = · δχ. · dx dy dx We can obtain another curve of the same character by plotting the lengths taken from a slide rule on the same base. This is the upper dotted curve in Fig. 38. The graduations of a slide rule are ruled proportional to the logarithms of the numbers engraved on the rule, so that addition and subtraction on the rule, which are easily performed mechanically by sliding one scale over the other, are equivalent to multiplication and division respectively. the ordinary small "Gravet" rule, log 10 is represented by 12'5 c.m. = 4'921 inches. 4'921 Hence this latter curve is = 2135 times as high 2°302 as the e curve, and its slope at the point (1,0) is 2*135 (§ 28). The result we have obtained may also be written log x + c. In this form it is extremely useful to the engineer in enabling him to find the work done by a gas (such as air) in expanding isothermally or at constant temperatures. This will be fully considered in the next chapters. § 42. D.C. OF e*. By inverting the curve of logarithms, as explained in § 23, we can prove a result of great importance. The curve P1 (Fig. 39) is the curve Y = log X. Rotate it about point O into the position dotted, and reflect on OY, and we get a curve whose equation might be written conformably with those of sin x and cos x, etc., y = log-1 x, or is the number whose logarithm is x. It is usual, however, to write the equation y = e*, for e* is obviously, from the definition, the number whose logarithm to base e is x. If the student cannot understand this, he is referred to any book on algebra which contains a chapter on logarithms. If this curve is differentiated graphically, the result will be a curve which is exactly in every respect like the primary curve. In other words, the peculiarity of this curve is that if the tangent at a point P be produced so as to meet OX in S, then, wherever P is on the curve, Sp will be exactly I inch, for the triangle SpP is evidently exactly equal and similarly situated to the triangle we should have drawn for the point P in differentiating the curve in the ordinary way. This is expressed by saying that the "subtangent" is constant, Sp being the subtangent. This result may be proved as follows:- An interesting property of this curve is that, if a series of abscissæ are taken in arithmetical progression, the corresponding ordinates are in geometrical progression. The student should prove this algebraically from the equation to the curve. The whole curve should be obtained by this method, by taking ordinates o 25 inch apart. The ratio of the ordinates will be e025, the value of which must be calculated, and the successive ordinates found geometrically by a construction similar to that of Fig. 3. All logarithms can be graphically obtained from this curve by measuring the abscissæ corresponding to an ordinate whose length = number whose log is required. The whole of the results in this chapter and the last must be thoroughly learnt off by heart. The student who wishes to proceed with the subject will save himself much time and annoyance by making himself perfectly familiar with them at the outset. It is not too much to say that one-half of the difficulty usually met by elementary students of the integral calculus is due to an imperfect knowledge of these few simple results. The student can best learn them by deducing them for himself once every day, and constantly picturing to himself the curves representing the functions and their differential coefficients. He thus obtains a practical and real familiarity with the functions, such as he could not get by studying the symbols only. Unless he is gifted with an exceptional memory, he will find even the few here collected difficult to remember otherwise than by understanding what they mean. The results should be as familiar forwards as they are back wards; e.g. he should know that dx = sec-1 x just as |