Adding we get AE (AB+BC) + BF (AB+ AC) = AEX AB+ AEX BC+BFX AB+BFX AC =AB (AEBF) + AB × EF AB(AE+EF+FB) = A B2. NOTE "ON THE HORIZONTAL THRUST OF EMBANK MENTS." BY F. W. BARDWELL, Professor of Mathematics in Antioch College, Yellow Springs, Ohio. IN an article "On the Horizontal Thrust of Embankments," in Vol. I., p. 175, I find what seems to me to be an error, which makes the principal formula defective, and therefore affects the conclusion. Without repeating the preliminary explanation, I will simply quote the portion which seems to be wanting in accuracy. "The variable forces, F' and Q, may be resolved into the components F' cos v, Q sin v, perpendicular to af, and F' sin v, Q cos v, parallel to af. The force, Fsin v, acting along af upwards, must, assisted by the friction due to the normal pressure on af, just equal the parallel and opposite force Q cos v; that is, (1) F" sin v+(F" cos v+Q sin v) f = Q cos v." But the normal pressure is Q sin only, and equation (1) should be F'sin v+Qf sin v=Q cos v. This corresponds to the formula usually given. I will only add that no allowance is made in the article referred to for the cohesion of particles. NOTE ON DOUBLE POSITION. BY REV. THOMAS HILL, President of Antioch College, Yellow Springs, Ohio. In the Cambridge Miscellany, Prof. PEIRCE expressed his regret that the old "Rule of False" had been omitted from the more recent elementary treatises on Arithmetic, published in this country. Since the date of that note, several Arithmetics have been published, containing the rule, introduced however, as it were, timidly, and in a corner of the book. Yet the fundamental idea of the rule of double position lies at the basis of every attempt of the human intellect to discover truth. We frame an hypothesis, compare it with facts, and note the amount of discrepancy; then, for the corrections of our hypothesis, assume that the errors of our results are in some measure proportional to the errors of our data. Now in mathematics, this gives a practical mode of attaining numerical results. Thus, let x be an unknown number, required to produce a fixed result, a. Let x and x" be two supposed values of x, which lead to the fixed results, a+e, and a +e. Now, according to the general assumption of proportionality of errors in results to those in the data, The identity of this formula with the following rule, given in DABOLL'S Arithmetic, is manifest. "Multiply the first position by the last error, and the last position by the first error. If the errors are alike, [that is, both results too great, or both results too small,] divide the difference of the products by the difference of the errors, and the quotient will be the answer. If the errors are unlike, divide the sum of the prod ucts by the sum of the errors, &c." The advantage of this form of the rule is the ease with which it disposes of the signs of the errors. If the problem is an equation. of the first degree, the answer is at once obtained with perfect accuracy; if not, the process must be repeated. For example: What is the side of a cube when the inches of superfices exceed the inches of solidity by 20? = Suppose a side of 2 inches. Then 6 (2)2-23-16 and 2016 = 4. Suppose a side of 3 inches. Then 6 (3)2 - 33 27 and 20 27 7. Hence, by the rule, the approximate side is (2 × 7+3×4) ÷ (7+4)=2.36. Taking, therefore, the two positions, 2.3 and 2.4, we have 6 (2.4) (2.4)3 20.74 and 20-20.74.74, = 6 (2.3)2 — (2.3)3 = 19.57 and 20 — 19.57.43. Then by the rule [(.74 × 2.3)+(.43 × 2.4)] ÷ (.43.74) Taking the new positions 2.336 and 2.337, 19.994; and 2018.994.006 6 (2.336)2 — (2.336)3 19.994; and 20 = = 6 (2.337)-(2.337) 20.0058; and 20-20.0058.0058. Hence, for a nearer approximation, the side equals + [(.006 × 2.337)+(.0058 × 2.336)] ÷ (.006+.0058)=2.336508. A simple rule, but requiring more care in the signs, consists in making the correction of the first position the unknown quantity. By the doctrine of proportions we readily obtain, from the given proportion, the following value of this correction, That is to say, Multiply the error of the first result by the difference of the positions, and divide by the difference of the results; the quotient is a correction to be added to the first position, if the first result shows it to be too small, and to be subtracted from the first position if the first result shows it to be too large. This rule we prefer to the old rule quoted from DABOLL, although it gives, of course, the same result. ON THE INDETERMINATE ANALYSIS. By Rev. A. D. WHEELER, Brunswick, Maine. [Continued from Page 25.] PROPOSITION VI. If a and b be prime to each other, the indeterminate equation, ax — by=c, is always possible; and will admit of an infinite number of positive integral solutions. -c+by DEMONSTRATION. Transferring and dividing, we have x= which has already been shown to be possible. PROP. V., Cor. 2. a Now as the remainders, resulting from the division of c+by by a, recur in periods, and the number of periods is unlimited, since y may have all possible values, it follows that the number of solutions must also be unlimited; that is, infinite. PROP. VII. If, in the equation ax + by=c, we call the least integral value of x, v; the next greater value of x, when the equation admits of more than one solution, will be vb, the next v+2b, and so on in Arithmetical progression. DEM. The least value of x being v, and d denoting the difference, we shall have v+d for the next greatest. Then ax + by becomes av byc in the first case, or y' = cav In the second case, it becomes avad± by"=c, or y" an integer. = d b (Ax. 2). Therefore is an integer. (PROP. III.) But this can be an integer only when d=b, d= 2b, &c., which was to be proved. PROP. VIII. The equation ax + byc is always possible for n solutions, when c>na b. DEM. Let cnab+r. ax+by=nab+r; or ax-nab――by+r; or (PROP. V. Cor. 1). Therefore the equation admits of at least one solution. Let xv for its first value; x=v+b for its second; x=v+ 26 for its third; and so on, (PROP. VII). Then we shall have x= v(n-1)b, for its nth value; and substituting this for x, we have av + (n−1) ab+by=nab+r; or Therefore the equation ax + by=c, will always admit of n solutions when c> nab. PROP. IX. The equation ax + by=c is impossible, in positive whole numbers, in the following cases. (1.) When a, or b, is prime to c, but not prime to each other. (2.) When c <a+b. (3.) When c = ab. ( (4.) When cab — (ax' —by'). DEM. Case 1. Since a and b are supposed to have a common factor which is not in c, it is obvious that one member of the equation is divisible by it, while the other is not. Case 2. The smallest integral value which can be given to x or y is 1. Giving to them this value, the equation becomes a + b = c. Consequently if c is less than a + b, the solution is impossible. |