GB878056A - Improvements in or relating to slide rules - Google Patents

Abstract

878,056. Slide rules. BOARDMAN, H., and THORNTON Ltd., A.G. Oct. 8, 1957 [July 18, 1956], No. 22180/56. Class 106 (1). A logarithmic slide rule bears special scales to facilitate the solution of quadratic equations. The quadratic equations x<SP>2</SP> Πbx + c = o and x<SP>2</SP> Πbx - c = o have related equations Q<SP>2</SP> ΠN s Q + 1 = O and Q<SP>2</SP> ΠN D Q 1 = O, where N s and N D = b#c. The expressions giving the roots of these equations involve one variable only (N S or N D ) and the slide rule, which has the usual A, B, C and D scales, is provided adjacent to the D scale with logarithmic scales of N S and N D so drawn with respect to the D scale that for any value of N S or N D within the range of the scales, the corresponding value for Q may be read on the D scale opposite the given value on the Ns or N D scale. In use, the value of Ns or N D is first found from the known values of band c using the D and B scales in the normal manner. Then using this value on the appropriate N S or N D scale, Q may be read, or by using a reciprocal square scale RB, which is preferably provided on the slide, one of the roots of the original equation which is given by x = Q#c may be found. The other root may then be found using the C and D scales. The special scales are used to find real roots only, and a conventional scale #(s) corresponding to a function #<SP>1</SP> - S<SP>2</SP> may be marked on the stock for use in evaluating complex roots. Special scales may also be provided for the solution of cubic equations. The cubic equation x<SP>3</SP> + ax<SP>2</SP> + bx + c = O has a related equation Z<SP>3</SP> - HZ + 1 = O, where H can be found in terms of a, band c. The expressions for the roots of this equation again involve only one variable (H) and consequently a scale can be constructed relating H and Z (read on the D scale). As shown, the scale is drawn in five ranges Hon and Hnm on the stock and, for convenience, Hpq, Hqr and Hrs on the rear of the slide. The latter three scales are read against an index line in a window on the rear of the stock, the corresponding Z values being read on the D scale against the end of the C scale. From the values of Z, the corresponding values of x may be found from an equation relating x, Z, a, band c. To facilitate the calculation of H, a cube scale F is provided on the stock. Specification 878,057 is referred to.