vision or by the method of indeterminate coefficients, art. 41, page 95, + (a+x)= a 2a2 3x3 -&c.+C. If we make x-o, the constant C=log a; consequently we shall have 1 (a+x) = la+ x x2 a 2a1 + 23 —&c.; and writing —≈ for x, we obtain 1 (a—x) = la — — 3a3 a % or r= -, by a+2 a a+z substitution we shall find l (a−x) = 2la—l (a+2) = la 2 (a+z)2=&c.; and therefore 1 (a+z) = la+ a + x + 2 ( a + z )2 + &c., a series converging with a degree of rapidity proportionate to the excess of a above z. For example/100=1 (99+1)=799+ 1 7 +2(100)2+&c. = 4.60517018; and 7 11=?(10+1) = / 10+ 100 + 11 duced into a series, gives y=x—x2x+x+xxx+&c. therefore y any arc, r its sine, or y arc sin x, we shall have = (1-xx); this expression expanded into a se √(1—2x)=2 (1−xx)—4; The logarithms found above are the hyperbolic or Naperean; these are directly converted into the common logarithms, by multiylying them by the constant factor 0.434294482. ries by the binomial theorem gives y=¿ (1+1-32+1.3 1.3.5 2.4 2.4.6 1 +&c.). Therefore y or arc sin x=x+. + + 2 3 2.4 5 2.4.6 7 +&c., a fluent to which it is not necessary to add a correction. Letx=1, T and call the semicircumference, and we shall obtain =1 74. These examples suffice to explain the preceding method. That which follows will merit attention. The formula flux (xy)=xy+yx, gives xy=fxy+fyx; therefore, in general fxy=xy-fyx, and if we denote by X any function of x, we shall in like manner have Xx=Xx-fxX. Call X=X' &; then by the same principle sâÑ, orƒX1 xi — X' xx — sxx X1. = 2 2 X = X" ; and we shall have ** X-X"-ƒ 2 2.3 Let again 2.3 Substituting these different values in the first expression, we shall we X 75. Let now y=m (a+x)-1, of which the fluent is y=(a+x)" ; shall have by the above theorem X= m (a + x)=-1, ÷=m (m—1) (a+x)"', —=m (m−1) (m—2) (a+x)"3, &c. I Therefore y, or (a+x) =C+m2 (a+x)=-1____ m (m—1) (a+x)m−1. 2 Let xo, and we shall m (m-1) 2 have Ca", and (a+x)" aTM +.mx (a+x)=-1 — x3 (a+x)TM-2+&c. Make a+x=z, and we shall obtain 2”—(≈—1)" m (m—1),22 m2 + &c. therefore +mxxm−1+ 2 These latter series will be found useful in extracting the roots of numbers. 76. To find the value of y=a*, take the fluxion, and we have (art. 18. page 444) y=a* xla. Therefore Xala, a l'a, =a* a, &c. which gives y, or a=C+a la-xx l2 a a2 + Pa __&c. 2 Let ro, and we shall have C=1, and 2.3 485 xx 12 a a2+&c.; dividing by a we shall obtain 1=a~+xla—xx l2 a 2 2 -&c.; and consequently if we suppose a positive, its odd powers must change their sign which renders the series all positive, and therefore xxl' a x3 13 a a=1+xla+" + +&c., as we already know. 2 2.3 77. Let now y be any arc, x its tangent, and we shall have y= 1+xx ; but as by making X = 1 we should obtain a very complicated series for the value of the arcy, we must somewhat modify the preceding method. 2 = 1 + xx + 3 (1 + xx)2 + Therefore in general (since tan A sin' y+&c.); or since sin A cos Asin 2A, 12.4 3.5 3.5.7 OF THE INTEGRATION OF LOGARITHMIC AND EXPONENTIAL FLUXIONS. 78. To integrate the logarithmic fluxion Xr lr, supposing X to be any function of x, let y=l, and ż=X, we shall then have ƒXile=ƒyż—yz—szj—le ƒ Xi—sz 2 Therefore the integral of the quantity proposed is reduced to that of Xr, and of. of ±ƒXi. Hence it may be found by the foregoing rules, if Xs does not contain any transcendental quantity. subject to no other exception than the case of n- 1. But then we |