roots.

117045

The following figures are intended to be a graphic explanation of the application of the method for the extraction of the higher Of the roots as far as found, r represents the last figure, R the portion preceding. A quantity at the beginning of a bent brace is to be multiplied by the multipliers along the brace, the upper for the true, and the lower for the trial divisors. For the fourth, fifth, and sixth roots, supplementary figures are shown, which are to be used for obtaining quantities, which when multiplied by proper powers of the figure of the root, found by a trial divisor, are added to the successive trial divisors to make the true divisor.

The expressions found by the several multiplications can be seen to be the proper expressions which are sought. The upper one being the true divisor, and the lower, the trial divisor.

For extracting the nth root, the following is general. The multiplicands are the several powers of R below n - 1 multiplied by the coëfficients of a binomial to the nth power. The upper expression at the end of the braces is the expansion of (R+r)" divested of its first term and divided by r. The lower expression is the expansion of (R+7)"1 multiplied by n. The upper set of multipliers are the several powers of r, in order below that of n-2 the higher one being repeated at last brace. The lower multipliers are the same powers of r as the upper ones multiplied by the digits in order. For the multipliers of a supplementary figure, the powers of r for the last term in each line agree in order with the number of the line commencing with the upper one: the coëfficients can be obtained by using the expression m being taken equal with the exponent of r of the multiplier. For the other multipliers, the exponent of r in each agrees with the formed columns in order, and the coëfficients of the columns are the same as the following differential series in respective position.

n-1

n-2

(n − 1) (n − 2........m

2, 3, 4, 5......m

Division is a process of evolving one factor, where another factor and their product are given. The method proposed is applicable, and its advantages are: that there will be no such thing as Long Division, the division by large numbers being as easy as dividing by a single figure; that fewer figures are used; and that the result is in its proper numerical position.

EXAMPLE.

Divide 64852 by 31.

2 6

31)6 4 8 5 2

2092

The divisor can be placed at the right as is the custom of the French.

RAILWAY TIME SERVICE. By P. H. DUDLEY, of New York, N. Y.

CORRECT time signals from various observatories have been daily sent over the telegraph wires of several railways for many years past, for the purpose of keeping all the clocks of a line alike. If the clock was in error the operator at each station must set it. Practically, the operators did not set their clocks if within a minute or two of correct time.

The problem was to arrange a practical system so that all of the clocks upon a line would be under the direct control of the Train Despatcher, though one or any number of miles away.

I have solved this problem by taking an ordinary clock with pendulum and weight, and arranging an electro magnetic apparatus, so that at given times the hands could be set. One of these clocks is

put in each station. By a three point switch, one side of which is connected to the clock, the other side to the sounder, and a wire to the relay, the clock or sounder may be put in circuit at pleasure. A few seconds before the time of sending the time signal as received from the Observatory, the Train Despatcher calls up his line to switch the clocks into circuit; and, when he receives the correct time signal he sends one dash to all the clocks, correcting them if either fast or slow. The clocks are now switched out, and the sounder put in, thus using existing telegraph wires only.

When this was successfully accomplished a demand was at once made for clocks beating seconds, to have the second hand corrected, at the same time of the minute hand, and for the past eighteen months, many clocks have been put in various railway stations, giving the public the benefit of Observatory time to the second. From many interviews with railway officials, I believe they will be ready to adopt a uniform standard of time for the entire country whenever the public demand it.

This question is so important, involving so many interests, it seems to me it should further engage the attention of the eminent men of this body, who have already done so much in this field, so that we may have the most practical solution of the question possible.

TITLES OF OTHER PAPERS READ IN SECTION A.

LECTURE EXPERIMENT FOR THE DIRECT DETERMINATION OF THE VEBy W. A. Anthony, of Ithaca, N. Y.

LOCITY OF sound.

A LECTURE EXPERIMENT, SHOWING THE MOVEMENT OF A HORIZONTAL CURRENT IN THE EARTH'S MAGNETIC FIELD. By W. A. Anthony, of Ithaca, N. Y.

A PROPOSED IMPROVEMENT IN THE CONSTRUCTION OF THE GRAMME ELECTRIC MACHINE. By W. A. Anthony, of Ithaca, N. Y.

A TABLE OF REMAINDERS OF 2" TO VARIOUS PRIME MODULI. By E. P. Austin, of Boston, Mass.

THE ASTRAL LANTERN. By F. H. Bailey, of Hillsdale, Mich.