Smithsonian Miscellaneous Collections, Volume 74, Issue 1

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Smithsonian Institution, 1922 - Science

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Page 76 - I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.
Page 5 - Every equation of an odd degree has at least one real root whose sign is opposite to that of its last term.
Page 62 - The cosine and sine of the sum of any number of angles in terms of the sine and cosine of the angles are given by the real and imaginary parts of cos (*i + xt +. . . . + xn) + 2 sin (xi + x.
Page 245 - I is encountered in its simplest form, not as the elliptic arc, but in the expression of the time in the pendulum motion of finite oscillation, unrestricted to the small invisible motion of elementary treatment. The compound pendulum, as of a clock, is replaced by its two equivalent particles, one at...
Page 5 - ... as follows : — No equation can have more positive roots than it has changes of sign from + to —, and from - to +, in the terms of its first member.
Page 4 - To transform the equation f(x) = о into one whose roots are the reciprocals of the roots of the given equation : Substitute 1/x for x and multiply by x".
Page 32 - OK is perpendicular to the plane zOz' drawn so that if Oz is vertical, and the projection of Oz' perpendicular to Oz is directed to the south, then OK is directed to the east. Angles z'Oz = 0, yVK = ф.
Page 6 - Xг are successively substituted in them. 1.266 Routh's rule for finding the number of roots whose real parts are positive. (Rigid Dynamics, Part II, Art. 297.) Arrange the coefficients in two rows: x...
Page 6 - X = + со 3 changes 2 changes i change Therefore there is one positive and one negative real root. If it can be seen that all the roots of any one of Sturm's functions are imaginary it is unnecessary to calculate any more of them after that one. If there are any multiple roots of the equation f(x) = о the series of Sturm's functions will terminate with fr, r < n.

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