The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . y given point in it, upto the point (£, 77), we have dp=da, or, integrating between the twopoints at which the radii of curvature are p2 and p„ we have p2 — ptr=<72 —0-!=: the arc intercepted between the radii of curvature. T
The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . y given point in it, upto the point (£, 77), we have dp=da, or, integrating between the twopoints at which the radii of curvature are p2 and p„ we have p2 — ptr=<72 —0-!=: the arc intercepted between the radii of curvature. That is(page 360), the excess of QQ over PP is the arc PQ. If, then, athread were placed in the position PP, and extended backwards in thedirection P Q to an indefinitely distant point, remaining on the evolutefrom P, and if this thread were unrolled, being always kept stretched, apencil at P would trace out the involute. Here, then, are, to allappearance, the two arbitrary constants of the involute to a givenevolute: we may take any point we please of the evolute ; that is, one ofits coordinates may be anything we please, the other being determinedby the equation; and at this point we may assign any length we pleaseon the tangent, to be the radius of curvature of the involute at the pointcorresponding to the one we have chosen on the evolute. Thus, if we. have an oval curve, and if we choose the point P as that at which theradius of curvature is PK, we have KAM for the involute (in part).But if the radius of curvature were PL, then LBN would be the in-volute. But we shall soon show that these two arbitrary constants areequivalent to one only, for we do not get more involutes by varying both,than we should do by varying one only. Thus the same involute whichwe get off the point P by assuming PK, we also obtain off the point Q byassuming QR. And we may evidently see that if the point A be given(which requires only one constant) the whole involute follows. The explanation o
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