Part 33

In one form of 10 in. rule, supplied by Mr. W. H. Harling, London, the body of the rule is made of well-seasoned cane, with the usual celluloid facings. The rule has a metal back, enabling the fit of the slide to be regulated. This backing extends the full length of the rule, openings about 1 in. long being provided at each end, enabling the scales on the back of the slide to be set with greater facility than is possible with the notched recesses usually adopted. The author has long endeavoured, but without success, to induce makers to fit windows of glass or celluloid in place of the notched recesses. This would allow the graduation of the S and T scales to be set more accurately, and enable both to be used at each end of the rule—an advantage in certain trigonometrical calculations. It would have the further advantage of permitting each alternate graduation of the evenly-divided or logarithm scale to be placed at opposite sides of one central line, enabling the reading to be made more accurately and conveniently.

Many special slide rules have lately been devised for determining the time necessary to perform various machine-tool operations and for analogous purposes, while attention has again been given to rules for calculating the weights of iron and steel bars, plates, etc.

THE DAVIS-STOKES FIELD GUNNERY SLIDE RULE.—This rule, which is adapted for calculations involved in “encounter” and “entrenched” field gunnery, is designed for the 18 pr. quick-firing gun. The upper and lower portions of the boxwood stock are united by a flexible centre of celluloid, thus providing grooves front and rear to receive boxwood slides. Each of the nineteen scales is marked with its name, and corresponding scales are coloured red or black. The front edge is bevelled and carries a scale of 1 in 20,000. The rule solves displacement problems, map angles of sight, changes of corrector and range corrections for changes in temperature, wind and barometer, etc. A special feature for displacement calculations is the provision of a 50 yd. sub-base angle scale, by which the apex angle is read at one setting.

THE DAVIS-MARTIN WIRELESS SLIDE RULE.—In wireless telegraphy it is frequently necessary to determine wave-length, capacity or self-induction when one or other of the factors of the equation, λ = 59·6√(LC) is unknown. The Davis-Martin wireless rule is designed to simplify such calculations. The upper scale in the stock (inductance) runs from 10,000 to 1,000,000; the adjacent scale on the slide (capacity) runs from 0·0001 to 0·01 but in the reverse direction. The lower scale on the stock (wave-length) runs from 100 to 1000, giving square roots of the upper scale; while on the lower edge of the scale are several arrows to suit the various denominations in which the wave-length and capacity may be expressed.

IMPROVED CURSORS.—In some slide-rule operations, notably in those involved in solving quadratic and cubic equations, it not infrequently happens that readings are obscured by the frame of the cursor. Frameless cursors have been introduced to obviate this defect. A piece of thick transparent celluloid is sometimes employed, but this is liable to become scratched in use. Fig. 37 shows a recent form of frameless glass cursor made by the Keuffel & Esser Company, Hoboken, N.J., which is satisfactory in every way.

Cursors having three hair lines are now fitted to some rules, the distance apart of the lines being equal to the interval 0·7854–1 on the A scale.

THE DAVIS-PLETTS SLIDE RULE.—In this rule a single log.-log. scale and its reciprocal scale are arranged opposite the ordinary upper log. scale. Thus, common logarithms can be read directly, while by taking advantage of the properties of characteristics and mantissas of common logarithms, the scale can be extended indefinitely. As 10 is the highest number on the log.-log. scale, it is carried down to within 0·025 of unity. The reading of log.-log. values above 10 is effected in a very simple manner. There is also a scale in the centre of the slide which, used in conjunction with the upper log. scale enables the natural logarithm of any number between 0·0001 and 10,000 to be read direct, while any number on the upper log. scale can be multiplied or divided by e^x if the latter is between these limits. On the back of the slide are scales for all circular and hyperbolic functions, these being used in conjunction with the upper log. scales.

THE CROMPTON-GALLAGHER BOILER EFFICIENCY CALCULATOR has a stock in the thickness of which is a slot admitting a chart which can be moved at right angles to the two separate slides. On the bevelled edge of one slide, the graduations are continued so as to read against curves on the chart, through an opening in the stock.

THE DAVIS-GRINSTED COMPLEX CALCULATOR.—This slide rule is of considerable service in connection with calculations involving the conversion of complex quantities from the form a + j b to the form R∠θ, and vice versa. The usual process of conversion necessitates repeated reference to trigonometrical tables, and is both tedious and time-taking. The Complex Calculator enables the conversions to be effected without reference to tables and with the minimum expenditure of time and labour.

The rule, which is about 16 in. long, has five scales. The upper one (A) is an ordinary logarithmic scale thrice-repeated. The adjacent scales on the slide comprise (1) a logarithmic scale of tangents (B) ranging from 0·1° to 45°, and (2) a logarithmic scale of secants (C) from 0° to 45°. The lower scales D and E are identical with the A scale, and are provided to enable multiplication, etc., to be performed without the need for a separate slide rule. Readings can be transferred from A to the lower scales by means of the cursor.

In using the rule to convert a + j b to R∠θ, the index (45°) of the B scale is set to the larger component and the cursor to the smaller component, on scale A. Then θ (or its complement if b is greater than a) is read on B under the cursor. The cursor is then set to θ on the C scale, and R is read on A under the cursor. The rule is made by Messrs. John Davis & Son, Limited, Derby.

THE SOLUTION OF ALGEBRAIC EQUATIONS.

The slide rule finds an interesting application in the solution of equations of the second and third degree; and although the process is essentially one of trial and error, it may often serve as an efficient substitute for the more laborious algebraic methods, particularly when the conditions of the problem or the operator’s knowledge of the theory of equations enables some idea to be obtained as to the character of the result sought. The principle may be thus briefly explained:—If 1 on C is set to x on D (Fig. 38), we find x(x) = x^2 on D under x on C. If, however, with the slide set as before, instead of reading under x, we read under x + m on C, the result on D will now be x(x + m) = x^2 + mx = q. Hence to solve the equation x^2 + mx − q = 0, we reverse the above process, and setting the cursor to q on D, we move the slide until the number on C under the cursor, and that on D under 1 on C, differ by m. It is obvious from the setting that the product of these numbers = q, and as their difference = m, they are seen to be the roots of the equation as required. For the equation x^2 − mx + q = 0, we require m to equal the sum of the roots. Hence, setting the cursor as before to q on D, we move the slide until the number on C under the cursor, and that on D under 1 on C, are together equal to m, these numbers being the roots sought. The alternative equations x^2 − mx − q = 0, and x^2 + mx + q = 0 are deducible from the others by changing the signs of the roots, and need not be further considered.

EX.—Find the roots of x^2 − 8x + 9 = 0.

Set the cursor to 9 on D, and move the slide to the right until when 6·64 is found under the cursor, 1·355 on D is under 1 on C. These numbers are the roots required.

The upper scales can of course be used; indeed, in general they are to be preferred.

EX.—Find the roots of x^2 + 12·8x + 39·4 = 0.

Set the cursor to 39·4 on A, and move the slide to the right until we read 7·65 on B under the cursor, and 5·15 on A over 1 on B. The roots are therefore −7·65 and −5.15.

With a little consideration of the relative value of the upper and lower scales, the student interested will readily perceive how equations of the third degree may be similarly resolved. The subject is not of sufficient general importance to warrant a detailed examination being made of the several expressions which can be dealt with in the manner suggested; but the author gives the following example as affording some indication of the adaptability of the method to practical calculations.

EX.—A hollow copper ball, 7·5 in. in diameter and 2 lb. in weight, floats in water. To what depth will it sink?

The water displaced = 27·7 × 2 = 55·4 cub. in. The cubic contents of the immersed segment will be (π)/(3)(3r x^2 − x^3), r being the radius and x the depth of immersion. Hence (π)/(3)(3r x^2 − x^3) = 55·4, and 11·25x^2 − x^3 = 52·9.

To solve this equation we place the cursor to 52·9 on A, and move the slide until the reading on D under 1 and that on B under the cursor together amount to 11·25. In this way find 2·45 on D under 1, with 8·8 on B under the cursor c, c, as a pair of values of which the sum is 11·25. Hence we conclude that x = 2·45 in. is the result sought.

With the rule thus set (Fig. 39) the student will note that the slide is displaced to the right by an amount which represents x on D, and therefore x^2 on A; while the length on B from 1 to the cursor line represents 11·25 − x. Hence the upper scale setting gives x^2(11·25 − x) = 11·25x^2 − x^3 = 52·9 as required.

When in doubt as to the method to be pursued in any given case, the student should work synthetically, building up a simple example of an analogous character to that under consideration, and so deducing the plan to be followed in the reverse process.

SCREW-CUTTING GEAR CALCULATIONS.

The slide rule has long found a useful application in connection with the gear calculations necessary in screw-cutting, helical gear-cutting, and spiral gear work.

SINGLE GEARS.—For simple cases of screw-cutting in the lathe it is only necessary to set the threads per inch to be cut to the threads per inch in the guide screw (or the pitch in inches in each case, if more convenient). Then any pair of coinciding values on the two scales will give possible pairs of wheels.

EX.—Find wheels to cut a screw of 1⅝ threads per inch with a guide screw of 2 threads per inch.