These formulas will next be tabulated; the degree of the given equation being m; the cubic type, Pr" + Qxn−1 + Rx”−2 + Sx2-3,

Here the first factors are formed alike from the exponents of P, R and Q, S, by the simple precept before given under the quadratic type. OUTLINE TABLE OF THE LIMITS.

=

= negative.

In respect to vacant terms, if either P or S is zero then f' or g' is zero; and by the table, the other g' or f' must equal or exceed 0.75 to indicate two imaginary roots. That is, simply, 0.75-f' When Q=0; we multiply the criterion by reducing the criterion to -4PR3-f(gPS). signs, this is negative, so that two roots are imaginary, as under the quadratic type. When P and R have unlike signs, - PR is positive, and the criterion is,

(C),

Q'R2 and then make Q=0,
When P and R have like

*This first value indicates generally two imaginary roots, otherwise two equal roots.

PHOTOGRAPHIC OBSERVATIONS OF THE

TRANSIT OF VENUS.

NAVAL

BY PROF. ASAPH HALL, NAVAL OBSERVATORY, WASHINGTON, D. C.

EDITOR ANALYST:

I was so busy during my absence, besides being cut off from the mail, that I failed to keep my promise to write you. Let me try to make amends now by giving you a short account of the photographic operations. The plan adopted by the American Commission for making photographs of the Transit of Venus is now pretty well understood. It consists simply of a lens of 38 to 40 feet focal length, so placed that its axis lies in the meridian, and is made horizontal by means of a spirit level. This lens is mounted on an iron pier firmly set in the ground, and south of this lens, at its focus, is fixed the plate holder, also mounted on an iron pier. The plate holder carries the sensitive plate, and directly in front of this plate a plumb line of fine silver wire. In front of the plumb line is firmly fixed to the plate holder a well made plate of clear glass, on which are finely etched two sets of straight lines, the lines being half an inch apart and the sets at right angles to each other. The distance from the north surface of the ruled glass plate to the surface of the sensitive plate is half an inch. On every photograph we have therefore traces of the ruled or etched lines and of the plumb line. North of the lens is placed a heliostat which moves a plane glass mirror so that the sun's rays are reflected through the lens to the sensitive plate. The whole apparatus is simple, it is firmly mounted, and it gives the means of determining accurately the polar coordinates of a point on the photograph with respect to an assumed point.

First we have a photograph of the sun, very nearly a circle four inches in diameter, and we note and record the instant of local time when the slide passes and admits the sun's rays and this photograph is made. For this instant we can compute for the known latitude of the station the position o the sun in the heavens, and hence the angular distance from the bottom or top of the photograph to the north point of the sun's image. The trace of the plumb line furnishes a known direction to which the measurement of the angles may be referred. If therefore we assume a point as the sun's center we can determine by measuring the photograph with a position micrometer the position angle of any other point, as the center of venus, with respect to the plumb line, or to a declination circle passing from the pole of the heavens through the center of the sun. This position angle can be found, I think, with much greater accuracy than it can be with an equatorial telescope, and here is one advantage of the American method. Secondly we must have

the means of converting any distance, say an inch, on the photograph very exactly into arc; so that if we measure the distance between the two points and convert it in to arc we have the second polar coordinate. This conversion can be made if we know the distance from the surface of the lens to the surface of the sensitive plate, just as we find the angular value of a micrometer screw by measuring the focal distance of the objective and counting the number of threads of the screw to an inch. The distance between the surfaces has been measured with an accuracy that seems much within the limits required. We have therefore the second polar coordinate and the solution is theoretically complete.

But in order to establish the usefulness of the photographic method it remains to be shown that the photographs can be made in distant places transported thousands of miles through great change of temperature, that the collodion film is subject to no contraction or change except such as can be fully accounted for in the measurement and calculation, and finally that their linear measurement can be made with the required accuracy. We We may easily get some idea of the accuracy required in the measurements of the photographs. The sun's diameter is about 32 minutes of arc, and the photographs beeng 4 inches in diameter we shall have 10% of an inch equal to 0.48, and Too of an iuch equal to 0.05. The photographs to be measured are the negatives on the glass plates, and on these Venus appears as a round vacant spot of an inch in diameter. As this spot is a symmetrical one, and generally well defined it is probable that the pointings on this spot can be made with sufficient accuracy. The difficulty will be in fixing the position of the sun's center. This can be done only by proceeding from the edge of the photograph, and the edge, or limb, of the sun is a very difficult thing to deal with, on account of its inequalities. Here perhaps some assistance may be derived from the systems of ruled lines, which are designed to control any contraction of the collodion film, and which may be used as a system of rectangular coordinates to which the centers of the sun and Venus may be referred.

We have now at the Naval Observatory a number of photographs from different stations, and shall probably soon know with what accuracy they can be measured.

SOLUTIONS OF PROBLEMS.

Solutions of problems in No. 2 have been received as follows:

From J. M. Arnold, 63; A. L. Baker, 61 & 65; W. W. Beman, 59, 60,

61, 63, 65 & 66; Marcus Baker, 59, 60, 61, 65, 66 & 67; Prof. P. E. Chase, 59, 60, 61 & 65; Geo. M. Day, 59, 60, 61, 63, 64, & 65; Dr. H. Eggers, 62 & 63; E. S. Farrow, 59, 60, 61, 63, 64, 65, 66 & 67; Henry Gunder, 59, 60, 61, 63, 64, 65 & 66; Wm. Hoover, 59, 60, 61, 63, 65 & 66; G. W. Hill, 67; Artemas Martin, 59, 60, 61, 63, 64, 65 & 66; J. B. Mott, 61; F. P. Matz, 59 & 60; E. P. Norton, 63; O. D. Oathout, 59, 60 & 65; L. Regan, 59, 60, 61, 62 & 63; E. B. Seitz, 59, 60, 61, 63, 64 & 65; Walter :Siverly, 59, 60, 61, 62 & 65; E. T. T., 59 & 60; J. S. Mays, 59 to 66.

59.-"Find a number, consisting of two places of figures, whose half squared will equal the number inverted."

SOLUTION AY PRES. E. T. TAPPAN, KENYON COLLEGE, GAMBIER, OHIO.

Problem 59 gives this equation [}(10x + y)]2 = 10y + x, . ' . y = 20 10x (400-396). Evidently the only integral value of x is 1. Then y = 8.

SOLUTION BY PROF. PLINY EARLE CHASE, PHILADELPHIA, PA.

==

Let 2x be the number sought. Then, in accordance with the Arabic system of numeration, a2 2x = x(x — 2) = 9y. Since x, the larger factor, can contain only one figure, if 2r is an integer y must be the smaller factor. .. x=9; y = 7; 2x=18. If 2x is decimal, its right-hand digit must be O, and the square of half the tenths digit must be one tenth of the digit. 2.2 = [Because the same digits that represent 2x also represent a2; therefore the excess of 9's contained in the sum of the digits is the same in both cases; consequently the excess of 9's in a2- 2x is zero. ... 2x is a multiple x2 . of 9, equal 9y say.—Ed.]

2x; x= .20; y

-=

1.8; 2x = .40.

x

SOLUTION BY PRES. E. T. TAPPAN.

62
X 32'

which is a whole number when

The given expression reduces to 32 is a measure of 62. Hence x may be 94, 63, 523, 471⁄2, &c.; 35, 33 &c.

61.-"What will be the value of each letter of the alphabet if the product of all but a is 1, all but b is 2, all bnt c is 3, and so on to all but z is 26.”

a=/(2.3.4....26)=11.59375; b=5.79688; c=3.86459; &c.

62.—“Given four lines in a plane: To inscribe a parallelogram within them with given direction of sides."

SOLUTION BY DR. H. EGGERS, MILWAUKEE, WISCONSIN.

The four given right lines may be named a, b, c, d, and the two given directions p and q.

d in §;

Now if

Take on the line a an arbitrary point P, draw from P a line parallel to P, till it intersects line 6 in Q; from Q draw a line parallel to q, intersecting line c in R; from R draw a line parallel to p, intersecting line from S draw a line parallel to q intersecting line a in a point M. the points P and M should coincide, the problem would be solved. But as this circumstance in general will not occur, take on the line a another arbitrary point P' and proceed exactly as above, till a second point M' results on line a, analogous to point M: Now take a third arbitrary point P" on line a and proceeding in the same manner, construct a third point M" on line a. The two systems of points on line a, viz; P, P', P" and M, M', M' will form two homographic systems of points on line a. Find now the two coincident points of these two systems of points, which can be done by one fixed arbitrary circle and the ruler alone. These two points may be X and Y; then every one will furnish one solution of the problem. For we have only to repeat the above trial construction, beginning with point X or Y, instead of the arbitrary point P; then the fourth line connecting d with a will pass through point X or Y respectively.

If The succession of the four lines a, b, c, d, is fixed, the number of solutions will be in general two. But if all possible combinations of the four given lines are admissible, we find 12 different combinations, consequently 12 X 224 solutions.

[Mr Siverly's solution of this question is by Analytical Geometry, and is brief and elegant. Mr. J. S. Hays, of Hodgenville, Ky., sent a very ingenious and elegant geometrical costruction and demonstration; his diagram is larger however than we can, consistently, admit to our pages; and we exact of correspondents, when diagrams are necessary to illustrate their subjects, that they be correctly drawn, suitable for the engraver to copy, and not to exceed three inches in their largest dimension; and always less when practicable.]