to very lengthy calculations. But though the common rule for finding the probability of error on a number of observations be as good as, or perhaps better than, any other at present known, I think it may be shewn that after all it has only a chance of being right, and is far from certainty in all cases. It may be a pretty good approximation when the number of observations is great, but when the number is small it seems somewhat dubious ; at least, when the number is a minimum it is palpably false, and it is not likely that it should become false per saltum. The rule commonly given, is to take the arithmetical mean of all the observations, and the difference between this and each observation; then squaring each of these differences, take the square root of their sum; which root divided by the number of observations gives the probability of error; the reciprocal of which gives the proportional weight due to each. I have long sought, but never met with, a demonstration of this rule. According to it, however great be the number of observations, if they differ among themselves by any quantity, however small, there will be a probability, however small, of error; and therefore the result must fall short of certainty. But if there be any number of observations agreeing among themselves, or even only one observation, there is no probability of error: so at least says the rule, whereas common sense says in the latter case the probability of error is very great, though we have no means of making a better of it. Hence I hold that the rule commonly given for finding the probability of error on a set of observations, though in general a pretty good practical rule, is not a mathematical truth: and I would not, on the faith of its being such, build a cumbrous computation to obtain a result not much, if at all better than that which may be obtained with one-tenth part of the labor: for such I believe would be the disproportion between the combination of these equations according to the method above indicated, and that by their weights as directed by writers on probabilities. As to the practical difference in the results by these two methods of adjustment, I cannot speak from actual trial; but I believe it would rarely exceed one or two hundredths of a second; and if we recollect that it is amazingly difficult with ordinary, or even with extraordinary instruments, to observe to within ten times the greatest of these quantities; and also that making adjustments by the sinal equations by means of Tables as in the final calculations to seven places of decimals, the difference of a single unit in the last place of a logarithmic sine corresponds at 45° to an angular difference of ~> "or about five hundredths of a second; and at 60° to ~- or about eight hundredths of a second; moreover, it is well known that in adding together several logarithms each of which is only approximately true in the last place, there can be no certainty that the sum will be true within one or two units in the last place; therefore the difference between the two methods of adjustment, (if I do not greatly err in estimating it,) may be considered as of no practical importance, being beyond the reach of the Tables. In regard to the exceedingly minute quantities which some of the continental observers used to profess themselves able to determine by means of the repeating circle, there is a very sensible remark by old Troughton, in a paper of the Astronomical Society, the substance of which I quote from memory, to the effect that whatever be the ability of the observer, or the construction of his instrument, he never would believe in the quantities deduced beyond such as were visible in the telescope. In fact, so long as observations have error at all, dispose of that error how we may, we cannot get rid of it so as to ensure certainty; the only advantage which the arrangement ultimately adopted can possess is, that of being a little better than a number of other arrangements equally possible, each of which is only somewhat less probable. N. B.—In the equations (B) (C) (S) the characters £], £» £3, t4> denote the excess on the respective triangles. SPHEROIDAL EXCESS. To find an expression for the Excess on a Spheroidal Triangle. By Captain Shobtbede, 1st Assistant Grand Trigonometrical Survey. It has been usual to consider the excess on a spheroidal as not differing sensibly from that on a spherical triangle of the same area as estimated by the mean radius of the earth, and this may generally be considered sufficient, for in the largest triangle ever observed, the difference is not a matter of observation. But as the two triangles have not absolutely the same excess, it may be worth while to ascertain the precise value of each, and thereby what would be the difference in any particular case. A solution may be obtained by means of the following principle given by Mr. Airey as resulting from Dalby's investigation, namely, that the excess on a spheroidal is the same as on a spherical triangle whose angular points have the same geographical latitudes and longitudes. The arcs being small, (as is always the case in practice), it is an assumption generally made and admitted, that the computations may be made by means of the radii of curvature at their middle points, which comes to the same thing as to reckon longitudes by a normal equal to the mean of all the normals at the middle points of the three sides, and latitudes by a radius of curvature equal to the mean of the meridional radii of curvature at the same points: or simply or at once, by the normal and meridional radius of curvature at the centre of gravity* of the triangle. The surface of the spheroidal triangle would coincide with that on a sphere if the differences of latitude were stretched out in the ratio of the meridional radius to the normal, or if the longitudinal differences were contracted in the inverse ratio. Hence the area, and also the excess on a spheroidal triangle would coincide with those on a sphere, if they be computed by a radius equal to a mean proportional between the normal and meridional radius (i. e. the greatest and least radii of curvature), at the middle of the triangle. This method (which is probably the simplest possible), is true as far at least as quantities of the 4th order, and in the present as well as prospective state of the arts and sciences, any thing farther may be thought an unnecessary refinement. If r denote the radius of the sphere, and A the area of a spherical triangle, the expression for the excess E reduced to its simplest form is A . A E = in terms of radius unity, or in seconds E" = and r3 r* sin V if the area be assumed as equal to that of a plane triangle having the same sides a, b, and contained angle C, the formula becomes • This expression may be objected to, as strictly the centre of gravity falls within the surface. It may be understood as an abbreviated expression, denoting the point of intersection of lines drawn from the angular points to bisect the opposite sides. E" = «Mr—r—TT.i in which the denominator being constant, and the 2 r2 sin 1 quantities in the numerator those which occur in the calculation of the triangle, the calculation of the excess becomes as simple as need be. To adapt this to the case of a spheroidal triangle, all that is necessary is to substitute for r* the expressions for the normal and meridional radius of curvature. Assuming a and B to denote the polar and equa torial radii, and f = -—, and X the latitude, the expression for a the normal is v = -—- —t' * )—, and for the meridional radius (i + r cos*X)$ «(! + «*) «'(!+**)* 7 = ("l + t«cos*A)-2 : heDCe thC F0<1UCt °f thC tw° 7 » = l+E«cos*A and this substituted for r* gives the excess on a speroidal triangle. a b sin C C1 + t* co^X)* a b sin C 1 + 2 t* Of the above expressions, the first three are identical and rigorous. The last is an approximation got by actually performing the operations indicated in that preceding it. It is however, sufficiently close for any ordinary purpose, as the quantities omitted affect only the 7th place of decimals. It appears from it that, when 2 cos*X = ~> 0r cos'X = ~> that is, in latitude 30°, the excess on the spheroidal triangle coincides with that computed by the mean radius. In lower latitudes the spheroidal excess is greater than the spherical, and only in latitudes higher than 30 is it less than on the sphere. To render this formula practically useful I have computed in the fol 1 1 R" lowing Table the value of the logarithm of — ;— , or for 6 2 y v sm 1" 2 y v every degree of latitude. I have assumed the values of the polar and equatorial radii of the earth as deduced from the comparison of the whole European with the Indian arc, as far as Kahanpur in latitude 24° 07, adapted to a ratio of axes 299 to 300. SPHEROIDAL EXCESS. As an example, suppose it were required to find the excess on a triangle having its middle point in latitude '24°, the sides a and 6 being 40 and 50 miles, and the contained angle C = 65°, the logarithms of o and b in feet being 5-3'246939 and 5-500785'2, the calculation would stand thus:— Tab Long for 44°, ... _. ... „. 0-57409 Long. Sin C = 65o, ... ... ... ... 9557*8 Long, a, ... ... ... ... ... 5-8M69 Long. 4, ... ... ... ... ... 5-50079 Long. E = 5."71S8, ... ... ... ... 0.75085 It is of no great importance to know the latitude of the middle of the triangle much within a degree, because the difference of the tabular logarithms for a degree never exceeds unit in the 4th place, which will scarcely ever give any sensible difference on the resulting excess. It will hereafter be shewn, that the error arising from assuming the area as equal to that of a plane triangle having the same sides and contained angle, is utterly insensible. |