## Smithsonian Mathematical Formulae and Tables of Elliptic Functions |

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angle Applying approximation C₁ coefficients column complete computed constant convergent coördinates correct corresponding cos x cos² cosh coth curl curvature curve decimal places definite Denoted derivatives determinant differences differential equation direction cosines elements elliptic equal Example expansion expressed Formula functions given gives integral interval limit linear method multiple negative normal obtained operator origin particular perpendicular plane polar positive radius respect result roots satisfied sin x sin² sinh SMITHSONIAN TABLES sn² solution solved substitution surface tangent tanh transformation values variable vector written wwwww y₁ ΙΟ ду

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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 - F(x) is the greatest common divisor of f(x) and f'(x), f(x)/F(x) = fi(x), and fi(x) will have no multiple roots. 1.260 An equation of odd degree, «, has at least one real root whose sign is opposite to that of an.

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 51 - D = o. 2.601 l, m, n are the direction cosines of the normal to the plane and p is the perpendicular distance from the origin upon the plane.

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 11 - The sign of any term is determined by the number of inversions from the second suffixes of the leading term, being positive if there is an even number of inversions and negative if there is an odd number of inversions. 1.341 The coefficient of...

Page 211 - Д' (p) is the first derivative of Д(/>) with respect to p, and the summation is .over all the roots, a. This solution reduces to « = o at / = o. Phil. Mag. 37, p. 407, 1919; Proceedings London Mathematical Society, 15, p. 401, 1916. 9.9 References to Bessel Functions. Nielsen: Handbuch der Theorie der Cylinder Funktionen. Leipzig, 1904. The notation and definitions given by Nielsen have been adopted in the present collection of formulae. The only difference is that Nielsen uses an upper index...

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.

Page 37 - ... vanishes and changes sign. 2.230 Eliminating x and y between the coordinates of the center of curvature (2.226) and the corresponding equations of the curve (2.200) gives the equation of the evolute of the curve — the locus of the center of curvature. A curve which has a given curve for evolute is called an involute of the given curve. 2.231 The envelope to a family of curves, i. F(x, y, a) = o, where a is a parameter, is obtained by eliminating a between (i) and dF *• ^7 = °da 2.232 If...