convenient to take at some distance from E. Through K draw KD perpendicular to S8, and make K D= K E tan / cot i. Join ED, and at the point N where it cuts the node draw a straight line perpendicular to the node. This line, if the angles and work are correct, will pass through P, because from similar triangles NP KD tan / , or PE tan = PN tani, as PE KE tan i above.

=

. NPPE.

Thus with the node, latitude and inclination given, the point P is found by the intersection of ED with S8. K may be any point on EL, but it is convenient to take it at some definite value of cot; for instance (our scale being the Earth's mean distance divided into 100 parts), K E tan / may be ico, 50, 25, &c. according to circumstances, as will be seen in the examples, post. When the projection has been found the developed orbit is easily obtained by making NQ = NP sec i.

The above method, which can be constructed very rapidly, offers a convenient plan for testing the accuracy of any given solution of the elements of the orbit of a comet, but for the purpose of the graphical working the process is as follows:-The direction of the node and the inclination of the orbit having been previously obtained, the method explained in this Section is used to bring the whole work together and to average the individual observations. After this has been done, and the different points so amended have been marked down, the work at this stage ought to be tried by the rule of the areas, and if it stands this test also, the small discrepancies which may still remain between the developement and the proper conic section (presumably a parabola) will be still further reduced by its comparison with that curve, and it will be seen what are the slight modifications which have to be made in the node or in the inclination, or in both, in order to reduce the outstanding errors. Whatever corrections are applied to these should be made by small instalments, and to each separately, and the effect noted down.

k k

SECTION 4. To find a Parabola having its Focus at 8 and which shall coincide with two Points of the Orbit.

In Fig. 233 let Q and P be the two points; usually the extremities of the developement. With the centre Q and at the distance Q S, describe the arc SNF; and with the centre P and at the distance PS, describe the arc SMG. The straight line which Fig. 233.

DIAGRAM FOR FINDING THE PERIHELION FROM GIVEN POINTS ON THE ORBIT.

is tangent to the two circles at M and N will be the directrix of the proper parabola, and from this all the other parts can be found. The curve when drawn may be conveniently applied on tracingpaper, keeping the focus on the place of the Sun and turning it about until it best fits all the points of the developement.

SECTION 5. The Measurement of the Areas in a Parabola. Fig. 233 may also be used to illustrate the exact rule for the measurement of the areas in a parabola. Let A be the vertex, A Sa, and let PH and QI be perpendiculars drawn from the principal axis A X.

If PSQ be the area of the space bounded between SP, SQ, and

the curve, then

PH (AH+3 AS) —QI (AI+3 A S) = PSQ.

6

SECTION 6. The Relations between the Time-intervals and the

Longitude Lines.

At the first opening of the enquiry, except the help given by the latitude numbers, as mentioned in Sect. 1, there is usually little to guide the student beyond the time-intervals and the longitude lines. It is important therefore to consider their relation to one another. Proportions founded on the time-intervals may generally be used as a useful first approximation unless the inclination of the orbit is very steep or there is a great change of direction in the path of the comet with respect to the node, between the different observations. As this may not unfrequently be the case, the remarks following should be taken into consideration.

In comparing the lengths of adjacent arcs in the orbit it can easily be shown that they are to each other inversely as the square root of the mean radius vector in each arc, and if the arcs are of limited extent are practically as the inverse square roots of the radii in the middle of each arc. This variation will of course affect the projection also, with which we primarily have to deal; but the arc in the projection also depends upon the general angle made with the node, which may frequently be taken without sensible error to be the angle which the chord of the arc makes with the node. Calling this angle, if measured on the orbit, a, or if on the projection, ß, we should find that if s be a small arc of the projection corresponding to S on the orbit, 8:S:: I sin2 a sin2i: 1; or 8: S:: 1:√1+ tan2i sin2 ß; the relation between a and ß being tan ẞ=tan a cos i.

It will be seen that when a or ẞ are small, and i not very great, the projection will have almost the same length as the original arc, and when these approach 90° the ratio of the projection to the original will be as 1 to sec i. Also it will be observed that when the inclination is very steep it produces great influence on these proportions.

It follows from the above considerations that although the length of an arc traversed in a given time increases or diminishes as the comet approaches or recedes from the Sun, yet when we compare the adjacent arcs of the projection, this tendency may be

greatly modified by the direction of its course with respect to the node. In the first approximations it is not desirable to try to calculate these effects minutely, although it will be useful to take some account of them when possible. But it may often be worth while to obtain a first approximation roughly, and from

this to deduce the effect produced by the causes above referred to, and then to rub out the first pencillings and proceed afresh with an amended table of the intervals. The diagram here given (see opposite, Fig. 234), which has been calculated from the formula

gives values of the length of a small are of the projection compared with the corresponding arc of the orbit.

If the orbit has been developed and the angular direction of its course a ascertained, ẞ is easily obtained from the relation tan ẞ= tan a cosi. As an example of this diagram, if ẞ= 30° and i= 45°, it will be found by the scale that s : S=9:10. Other values can be found by interpolation.

SECTION 7. Checks available, derived from certain properties of

Parabolic Orbits.

When the elements of a comet have been approximately ascertained, a very useful check may be employed (confining our attention to parabolic orbits) from a consideration of the fact that the velocity of a comet in such an orbit at perihelion is to that of a planet moving in a circular orbit at the same distance as √2 to 1.

The sine of the daily arc traversed by such a planet at Perihelion would be 1.7213 of our scale. In the comet at the same distance it would be 2.43302, and for any other distance this number must be divided by the square root of the distance.

It will often be useful to remember this principle at a preliminary stage, when a consideration of it may help to point out the distance at which the first approaches should be commenced. SECTION 8. Examples of the Graphical Process.

The first example to be given is that of Schäberle's comet of 1881 (iv). Observations on 5 days will be considered. It is proposed to find the elements of the orbit from the first 4 and then to try them on the 5th for a test and final correction.

c The motion of the earth in its orbit, although not quite circular, may without serious error be used in this comparison.

It will require very little calculation, as it has necessarily been laid down graphically in the course of the work.