NEW PLANETARY NEBULE. By EDWARD C. PICKERING, of Cambridge, Mass.

[ABSTRACT.]

DURING the past year, a search has been made for new planetary nebula with the large telescope of the Harvard College Observatory. A direct vision prism, placed between the objective and eyepiece, converts the image of a star into a colored line of light. A planetary nebula, on the other hand, being nearly monochromatic, appears as a point of light, and is distinguished at a glance from a star. Many thousand stars may be quickly examined in this way, and a single nebula selected from among them. Two new nebula have thus been found, having the positions, R.A. 18h 25m 10', Dec., 25° 13', and R.A. 18h 4m 198, Dec., 28° 12'. Either of these objects would be mistaken for a star if examined in the usual way. On the evening of August 28, an object was found with a very singular spectrum. The light consists mainly of two bright bands, one in the yellow, a little more refrangible than the D line, the other in the green or blue. The approximate wave lengths are from 5,800 to 5,830, and from 4,670 to 4,730. There is also a faint continuous spectrum. This object cannot in other ways be distinguished from an ordinary star, and in fact, has been observed as one, by Argelander, and at the Washington Observatory, about thirty years ago. It does not seem to have changed since then in position or magnitude. It is designated as Eltzen, 17681, and its position for 1880 is, R.A. 18h 1m 17, Dec., 21° 16'. The comparative faintness of the continuous spectrum makes it more nearly resemble a planetary nebula than a star, but since the bands are differently placed, the material must be different.

NEW METHOD FOR FINDING THE NUMERICAL ROOTS OF EQUATIONS BELOW THE FOURTH Degree. By JAMES D. Warner,

THIS method is new for complete equations, and for those cubic equations where x appears in more than one term. For others it is not wholly new, but the arrangement is claimed as an improvement.

For finding the roots of the incomplete quadratic, the following rule is given, viz.:

Under the number write the figures of the root, commencing under the first figure of the last half, or of the larger latter part when divided as nearly as possible into two equal portions. Having ascertained the first figure of the root, take its square from the number, writing all of the different figures of the remainder over the number, in their proper numerical places.1 Under the figure of the root, write its double, and with this as a trial divisor, find the next figure of the root. The square of this, together with the product of itself into the trial divisor, taken from the remainder, gives the next remainder. Adding the double of the last found figure of the root, to the line of the double of the root previously found, will give a new trial divisor, by which the next figure of the root can be ascertained. The same process pursued with this and each succeeding figure found, will finally evolve the whole

roct.

EXAMPLE.

x2=531441: or extraction of the square root of 531441,

1 430

531441

729

144

Observe, that when any figure in the lines of remainders has been used, and there is no figure over it, or over a succeeding figure, to cancel it, by either striking a line through it or over it; also, whenever in doubling the root, there is anything to carry, to put it as a subscript to the previous figure, and use the sum of the two thereafter. Any remainder can be set off, by a long cancelling line through or over the next preceding figure on the upper

In subtracting, add a unit to amount to be carried when the unit figure of the subtrahend exceeds the figure from which it is subtracted.

line of the remainders, or a dash with the remainder written after it, may be used.

The same method answers for finding the roots of the complete quadratic, by adding in the coefficient of the first power of x to the first trial divisor.

If the coefficient of 22 is other than unity, multiply each figure of the root by it, before every multiplication of the figures of the

root.

EXAMPLE. 2x2+3x=6.

5

12862
6.

1.138

526

742

5

A nearly similar method is applicable for finding the roots of 23a, or extracting the cube root.

RULE.

In extracting the cube root, the first figure of the root is placed under the first figure of the last part of the number, when the number of the figures have been divided as nearly as possible into three equal parts. The first line of figures under the root is three times the root, and is formed continuously from the root as found, just before obtaining the true divisor, and may be desig

nated, for convenience, the line of triple products. The lines under the line of triple products are successive trial and true divisors. A trial divisor is always found by multiplying by the last found figure of the root into its double and that part of the line of triple products previously found, and adding the sum of the two products to the last true divisor. A true divisor is found by multiplying the last found figure of the root and that part of the line of triple products already found, by the trial figure of the root, and adding the products of the last trial divisor. This last sum, if found to be correct after testing, is to be multiplied by the found figure of the root, and the product taken from the remainder will give the next remainder.

When a cubic equation has been divested of its second term, and the coefficient of x3 has been reduced to unity, the same method is applicable by adding the coefficient of the first power of x, to the first trial divisor.

EXAMPLE.

x3-12x = 28.
1
667

129302

28.

4.3 0213

129

36

3969

4347

9

The complete Cubic Equation can also be solved, by observing, that before each multiplication of a figure of the root, it must be multiplied by the coefficient of 23, and the coefficient of x2 must be added to the line of triple products.

A similar method may be used for finding the roots of equations of a higher degree than the third, but as it is too complicated for ordinary mental operation, Horner's method is preferable; but it can be applied to the extraction of the higher roots, with a little subsidiary work.

Subsidiary work, to assist the mental operation, with expla

nation (see graphic explanation).