Ai + B2 + Dj — 180 = £l B, + C2 + D2 — 180 = gf f Triangular equations. (j3) Ct + A2 + D, - 180 = f3 V As + B,+ C,-180-ei + 8j + fj Alao Di = 360 — D2 — D3 and substituting the values of D2 and D3 from the triangular equations = 360 + Bi + C2 — 180 — tj + Ci + A2 — 180 — t3 = Bi + C3 + A2 — e2 — ej .-. Bi + Cj + At — D4 = £ + t ) These may be term(8) p, + A, + B2 - D2 = H + H t^Cj^ej^ Ai + Bj + C2 — Uj = (J + £j J of theangles concerned. If we examine in detail, these angular equations, we shall find that in both cases, besides the totopartial equations, there are two separate eqaations of condition, whereby to determine the error on each of the 12 angles in the figure. As these, with the exception perhaps of the Vertical and Cuneal equations contain nothing new, I Bhall proceed to the investigation of the sinal equations, which indeed is the main object of the present communication. These I shall give in the form, in which they first occurred to me. Prop. I.—In a quadrilateral the product of the sine of any whole angle, and the sines of the two consecutive left hand angles going round by the left, is equal to the product of the sine of the opposite whole angle, and the sines of the two corresponding right hand angles returning by the right. Sin Aj sin Bi sin Ci = sin Cj sin Bj sin A2 These (for reaSin Bj sin Ci sin Di = sin D3 sin C2 sin B2 f sons afterwards ^ Sin Cs sin Di sin Ai = sin A3 sin D2 sin Ct t to be shewn) may Sin D3 sin Ai sin Bi = sin Bs sin A2 sin D2 J be termed external alternate equations. Prop. II.—In a quadrilateral, the continued product of the sines of two adjacent whole angles, and the sines of the angles between the diagonals, and the opposite sides, is equal to the continued product of the two pairs of opposite angles. ^ Sin A3 sin B3 sin Ci sin D2 = sin C3 sin D3 sin Ai sin B2 Sin B3 sin Cs sin Di sin A2 = sin D3 sin A3 sin Bi sin C2 These may be termed opposite alternate equations. Also the known property, that the product of the sines of all the left hand angles is equal to that of the sines of all the right hand angles. (G) Sin Ai sin Bi sin Ci sin Di = sin Aj sin B2 sin Ci sin D2 which may be termed the internal alternate equation. This result might have been deduced from (E), by dividing any of the equations by another as they stand: thus :— Sin Bs • sin Ci • sin Di _ sin Ds ' sin Ct" sin Bt and rejecting Sin Cs • sin Di " sin Ai sin As ' sin Di ' sin Ct commonfactors. Sin Bs sin Ct _ sin Ds sin Bt Sin Cg sin Ai sin As sin Dt .*. Sin As • sin Bj • sin Ci • sin Dt = sin Cs ' sin Dj • sin Ai ■ sin Bj Again multiplying vertically all the equations, (E) rejecting common factors, and taking the square root, we have Sin Ai • sin Bi • Ci sin Di = sin At ' sin Bt " sin Ct ■ sin Dt ..(g) In like manner in the 2d figure • we have the following equations: Sin As ■ sin Bi • sin Ds = sin Bs - sin At * sin Dt } Which may (H) Sin Bs • sin Ci ■ sin Di = sin Cs ■ sin B, • sin Ds ^ter^ln^d Sin Cs • sin Ai • sin Dj = sin As 1 sin Ct ' sin Di Jternate. Also Sin Aj • sin Bi ' sin Ci 1 sin Di = sin Cs - sin Bt • sin At - sin Dt (I) Sin Bs " sin Ci ■ sin Ai ■ sin Dt = sin As " sin Ct 1 sin Bt ' sin Ds Sin Cs • sin Ai • sin Bi • sin Ds = sin Bi ■ sin A ■ sin Ct ' sin Dt which may be termed medial alternate. And (K) Sin Ai • sin Bi ■ sin Ci = sin At' sin Bt' sin Ct which may be termed internal alternate. Demonstration. _ _ . _ sin At BC=AC9i^ = CD^AC!|aA-S!^ sin As _ sin At sin Dt sin B5 sin Ds sin Bi .". Sin As sin Bi sin Ds = sin Bs sin At sin Dt fh) Multiplying vertically pairs of (H), and rejecting common factors, we have the equations fij and multiplying vertically the whole of (H) in like manner, we have the equation (k) I found at first some difficulty in trying to express in a convenient form of words, the properties in the equations (H) and (I); but after some consideration it appeared that they were included in the expressions for (E) and (F) : for if the point D in fig. I. be conceived to move along the line B D till it comes within the line A C, the quadrilateral with its diagonals is transformed into the triangle with its radial lines. .The figure may now be considered as a quadrilateral with a reentrant angle, in which case the angles Ai and As exchange their designations, At remaining unchanged. The analogy of these figures may be otherwise apprehended by considering them as the perspective representations of a tetrahedron; which is a quadrilateral with its diagonals, when the apex is projected between the exterior angles of the base; and is a triangle with its radial lines, when the apex is projected within the base, or within the vertical angles formed by the sides of the base produced. These equations hold good in spherical as well as in plain figures, the only change in the demonstration being to substitute the sines of the spherical Bides for the plane's sides as above. ,. The equations marked (G) and (K) are obviously only particular cases of a more general property given by W. Davies in his Supplement to the spherical part of Young's Trigonometry, and there said to be due to Professor Lowry. Lowry's Theorem is this: " If great circles be drawn from the angular points of any spherical polygon to a point on the surface of the sphere, the product of the sines of the alternate angles will be equal." This theorem applies of course in plane as well as in spherical polygons, and it is not unlikely, that if we substitute lines of shortest distance (including at once both straight lines and great circles), it may be found to apply on a spheroidal, as well as on a spherical or a plane surface. On farther consideration I find that the equations (E) and (H) are also included in Lowry's theorem. In fig. I. D being the point to which lines of shortest distance are drawn from the angles of the polygon A B C Lowry's theorem gives at once the first of the equations (E) and taking successively the points A B and C in like manner, the other equations are evolved. Likewise on fig. 2 if C be the point to which the lines are drawn from the angles of the polygon A B D, we have the first of the equations (H) and the others by taking successively A and B as the point of drawing to (i. e. attraction in its primary sense). The sinal equations furnish in each case four equations of condition for each of the 12 angles concerned. I do not see how any of them can fairly be omitted: for although any one of them may involve all the others then the angles are free from error, such is not necessarily the case when the angles, as happens in fact, are mixed up with errors we know not how. I know not of any way by which a fair judgment can be Wed as to the goodness or badness of observations, besides that resulting from the amount of minimum alteration required to make the whole consistent among themselves. It is quite possible, and will generally happen, that in every one of the above equations taken singly, the errors will be so mixed up in two or more of the quantities concerned as in a greater or less degree to destroy the effect of each, which errors will become sufficiently apparent when the quantities are otherwise combined. Using the whole of the equations, if the correction for any one quantity retains the same signs throughout, while on another quantity the correction is in a great measure destroyed, being sometimes + and sometimes —, we may fairly infer that in the former case the observed quantity is erroneous, and in the latter that it approaches to its true value; the errors being in proportion to the algebraic sum of all the corrections. The Binal and angular equations of figure being quite independent of each other, I am not aware of any reason for preferring the one set of them to the other; it appears to me that both ought to be taken into account simultaneously, giving equal weight to the mean error as found from each set. By any other method, the ultimate corrections will depend on the arbitrary order, in which the equations may have been applied. It may, however, be expedient to apply the totopartial equations, which are independent of figure, after having taken the mean of the others. The practical use of these equations in the method above sketched, when we retain only what is necessary, though still somewhat long, is by no means very difficult. The most convenient way would be to take the sums of the effective probabilities, and the sums of the errors, and get the correction by common Rule of Three, by the help of a sliding rule. Writers on the doctrine of probabilities direct that when several independent quantities occur, they should be combined according to their weights, or inversely as their probabilities of error, as found by the common rule. This applied to each of the above equations would give rise * F |