Elements of analysis as applied to the mechanics of engineering and machinery . 15,and, finally, from the last formula:nat. log. 10 = nat. log. (8 -f 2) 16+2 ^ ^ V16 + 27 ^ J = nat. log. 8 -{- 2 = 2,0794415 + 0,2231436 = 2, can also put nat. log. 2 = nat. log. 1 + 2 [^^-^ + i (j-i-^J+ .. •] = 2 ( i + 4 • -|- + i • ]iH- • • •) = 0,693UV, further, AuT. ELEMENTS OF ANALYSIS. not. log. 5=naf. log. (4 -f 1) = 2 naf. Jog. 2 -f 2 \Jj4--l . ^H J, and lastly, nat. log. 10 = nat. Jog. 2 -L log. 5. (Comp. Art. 19.) Art. 24. The trigonometric and circular functions^ whose differ-ential
Elements of analysis as applied to the mechanics of engineering and machinery . 15,and, finally, from the last formula:nat. log. 10 = nat. log. (8 -f 2) 16+2 ^ ^ V16 + 27 ^ J = nat. log. 8 -{- 2 = 2,0794415 + 0,2231436 = 2, can also put nat. log. 2 = nat. log. 1 + 2 [^^-^ + i (j-i-^J+ .. •] = 2 ( i + 4 • -|- + i • ]iH- • • •) = 0,693UV, further, AuT. ELEMENTS OF ANALYSIS. not. log. 5=naf. log. (4 -f 1) = 2 naf. Jog. 2 -f 2 \Jj4--l . ^H J, and lastly, nat. log. 10 = nat. Jog. 2 -L log. 5. (Comp. Art. 19.) Art. 24. The trigonometric and circular functions^ whose differ-entials are likewise to be found in the following, are also of prac-tical importance. The function of sine y = sin. x gives, for 5? = 0, ^ = 0, TT _ 3,1416 for X = 0,t854 ...,y = V^ = 0,tO i 1, for X = y, y = 1, for x = 7:,y = 0, ioY x^%-^y = — 1, for a; = 2r, ^ = 0, &c.;hence, if we take x as abscissa J. 0, and y as corresponding ordi-nate OF^ we obtain the meandering curve (APBtz C2-), Fig. 33,which may be extended indefinitely on both sides of A. Fis;. 33. T K. -Y EL H N The function of cosine y = cos. x gives, for a? = 0, y = 1, for x = -J^ y = Vh foi^ X = ^, y = 0, foT X = -^ y ^ — l^ foT X = ^ tt,?/ =: 0, for ^ = 2 -, ?/ = 1, &c.; therefore, precisel} the same mean- derina: line (+ ^i^i - j which corresponds to the function of sine, corresponds also to the function of cosine; but it is, on theaxis of abscissas, by ^ - = 1,5T08 . further before or behind thecurve of the sine. The curves, however, which correspond to the and co-tangential functions y =^ tang, x and y = cotang. x^ are of entirely PjC, ELEMENTS OF ANALYSIS. [Art. 25. different form. If, in y = tang, x, we put a? ^ 0, |- -, i -, we obtain?/ ^ 0, 1, oo; and lience, a curve (A QE) which approaches nearerand nearer to a line running parallel with the axis of ordinates A Y and passing through the point I -^ ) of the axis of abscissas A X, but which never reaches it. If, further, we take x =
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