And by the elements answering to the date June 12, 1836. M 319 13 1.8 π= 250 7 56.4 L = M+π= 209 20 58.2 ON THE DEPENDENCE OF NAPIER'S RULES. By Rev. ANTHONY VALLAS, Phil. Dr., New Orleans, La. THE five parts of a right-angled spherical triangle being placed on the circumference of a circle, NAPIER'S Rules are as follows: RULE I. The sine of the middle part equals the product of the cosines of the opposite parts. RULE II. The sine of the middle part is equal to the product of the tangents of the adjacent parts. It must be remembered that, instead of the hypothenuse and the two acute angles, their complements are used, the right angle not being counted among the parts. That the second of these rules may be deduced from the first has been shown by Mr. SAFFORD, in No. I. Vol. I. We are going to show the same in the following way. Apply the first rule to the three parts a, a', ', and then to a, a', w. Eliminate one of the common parts, for instance a, and the result, properly transformed, will give the second rule. From sin2 tan2 a' tan2 o ... sin o'tan a' tan w; or NAPIER'S Rule II. In order to find the second rule from the first, take the three adjacent parts a, μ, a', and μ, a', a', equations and eliminate u from the two Replacing the tangents by their equivalents in cosines, we have ON A TRANSFORMATION OF THE DERIVATIVE OF ANY POWER OF A VARIABLE. By Rev. GEORGE CLINTON WHITLOCK, LL.D., Professor of Natural Sciences in the University of Victoria College, Coburg, Canada West. THEOREM. Any power of a variable, regarded as a factor of any power of the same, may be carried from under the sign of derivation, by adding algebraically the product formed on multiplying its exponent into the variable having an exponent less by unity than that of the undecomposed power; whatever n and r may be, plus or minus, whole or fractional, or imaginary. DEMONSTRATION. From the laws of exponents in general, whatever n may be, plus or minus, whole or fractional, or imaginary, we have ··· (x + h)" = (x + h) (x + h)"—1, by changing x into (x + h); consequently, subtracting member by member and separating the factors x, h, (x + h)" — x" = x [(x + h)n−1 — x2-1] + h (x + h)”−1, therefore, passing to the limit, that is, taking the values of the terms the derivative of " is equal to x times the derivative of 2-1 plus ; or a factor z of any power 2 may be carried from under the sign D, of derivation, by adding a′′-1. Therefore by (1) D (xn−1) = x D (x3-2) + 2o-2, and by replacing D (2-1) by this value, (1) becomes D(x") = x2 D (xn−2) + 2 xn−1; and in like manner this form may be changed into and in general, the same operation being repeated r times, we have (2) D(x") = x D (2~) + r 2′′−1; i. e. any number, r, of factors, x, may be carried from under the sign D by adding r2"-1. (3) From (2), dividing by x', we have x ̄D (x") = D(2”—') + r 2”--1, or D (x"~') = x ̄ D (2′′) — r 2”—1, or putting n—r=m, which may be any number whatever, since n (4) 1 we have the same rule as Again (2, 3), p and r being any whole numbers, plus or minus, a factor D(x) = x2* D (2a− r)+græ′′−1; i. e. being separated, a power of this factor, (x) = x2, will be separated by multiplying r by g. It follows that (5) D (x") = x2; D (x*") + "; since by the last, multiplying by, we have which, as necessary and sufficient, accords with (2). Therefore (2), (3), (5), the theorem is proved, since whatever is found for real exponents must be extended to those that are imaginary. SCHOLIUM. The deduction of (3) and (5) from (2) is in accordance with the general theory of exponents, as indicating, whether plus or minus, whole or fractional, or imaginary, the number of times the quantity to which they are affixed, must be taken as a factor, or so regarded, in executing any operation. Thus a is x taken r times as a factor, and theory requires that x be regarded as z taken — 1 1 times as factor, and so x-is x taken times as factor, and z = (x)′ is 1 taken r times as factor, or x taken r or times as factor. It is true that here, as elsewhere, we have to extend the signification of the phrase, “number of times," that is, "number" must be regarded, not only as continuous (of any magnitude between integers), but as plus, or minus, or imaginary. And it is by this extension, precisely, that ANALYSIS has its power, or better, an existence. COROLLARY.-Whatever n may be, from the theorem we have the derivative of any power of a variable is found by diminishing the exponent by unity and multiplying by the original exponent. |