Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/180

Rh -with a similar expressioii for the relation between I) and c. -cated in the diagram (fig. 5); suppose 168 each angle. is made, under siniilar t'il'Cll1llstilllCt‘S, of any measurable iiiagiiitiule —as. an angle —the weight of the result is equal to half the square of the number of observations divided by the sum of the squares of the ditfereiiees ot the individual ineasures from the mean of all. Now let Ii, A‘, l be the weights of the three measured angles, and let .1‘, y, : be the corrections which should be applied to them. “'1: know that :z‘+ 3/ + : 1 e=O; and the theory of probabilities teaches us that the most probable values are those wliieh inake /ta.-'3+k_I/'-'+l;'-‘ a minimum. llere we arrive :it a simple deﬁnite problcin, the result of which is lu:=l'_i/= 1:, showing that e has to be divided into three parts which shall be proportional to the reciprocals of the weights of the corresponding angles. In what follows we shall, for siiiipli- city, suppose. the weights of the observed angles to be all equal. Suppose now that A, I’), C are the three angles of a triangle, and that the observed values are .+e,, B+e.,, +c3; then, although
 * 1, c2, 03, the errors of observation, are unknown, yet byaddiiig up the

observed angles and finding that the sum is in excess of the truth by a small quantity 0, we get e,+eg+r3=c. Now, according to the last proposition, if we suppose the angles to be equally well observed, we have to subtract §e from each of the observed values, which thus become A + §e1 — §,-._ — ;',c3, B — so, + §r’.3 — 54,, C— §,t,', — §,r2+ §"3. Then to obtain (6 and b by calculation from the known side 1‘, we have (1 Sin (C — 5'31’, -— :—'5(’2 + §r3)=c sin (A -1-gr, — :1,e,, -— :1,:*,,), Put a, B, -y for the eotangents of A, 1}, C‘, then the errors of the computed values of a and b are e.'pi-essed tln1s— 5~t=§'¢{0i( 9a+1l+’-.»(—a+1)+c;i(-a—‘27)} 8’)=;1,b{¢'l(—B+7)+c._,(138+-y)+c3(—B—2-y)}' Now these actual errors must remain unknown; but we here make use of the following theorem, proved in the doc- trine ef proba.bilities. The probable error of a quantity which is a function of several independently observed ele- ments is equal to the square root of the sum of the squ-ires of the probable errors that would arise from each of the observed elements taken singly. X ow suppose that each angle in a triangle has a probable error c, then we replace cl, 6’._,, e,,_ by e, and adding up the squares of the coefficients find for the probable error of (1, i -,1,—ae ‘/6 /(a.9+a.y+ y‘-’), and for that of b, :1: ‘£116 JG /(/3'3 +,8y+ 37-’). Suppose the triangle equilateral, each side eight J miles, and the probable error of an observed angle 0"'3 ; then the probable error of either of the computed sides will be found to be 0'60 inches. Take a chain of triangles as indi- all the angles measured, and that the sides MN, HJ are nieisured bases; it is required to iiivestigate the iieces- sary corrections to the observed angles in order not only that the sum of the three angles of each triangle fulfil the necess iry condition, but that the length of IIJ, calculated from that of MN, shall agree with the measured length. Let X1, Y,, Z,, &c., be the angles as observed, 1-,, 3/,, 2,, &c_, the required corrections; then each triangle on adding up the angles gives an equation 9',+y,+:,+e,=0. Let the coriected angles be X‘=X+.r, 1‘==Y+_1/, &c., then ll.ly sin X: sin X‘, sin X}, sin X} M.'=sin 3'; sin Y; sin YT-s.iii Y: =sin 3:, sip sin X5, sin .:_,(1 +20. sin 3, sin )2 sin ‘I3 sin 3,, Let a,,'B,, 7,,. . . be the cotangents of the angles, so that sin .‘ =siii h(_1 + at‘), then 7!. in this last equation is easily seen to be the right hand member of the equation f= “rri ’ Bi?/1+ ‘1-.-5"-.r" B-.-3’-2+ - - Here f is known nuineria-all_v, for the ratio of the measured bases is known, and the produu-.t of the ratios of the siiies of the observed angles is known by coiiipiitatioii. The most probable values of 27:], 3/1, :1,. . . . are tl10SC_'lI'll‘ll make the sum Ef-‘I52-i 3/" +.:'-’) a iiiiiii- muin, or, as we may write it, ¢=3(~’«"1+3/"’+(‘3+-7”‘-‘Z/)2) GrEUl)ESY do this we must be guidet by the ireiglit of thc deterininations of | a iiiiiiiiiiuin. When a series of direct and independent observations I the corrections. Tl1is,.'1iid the previous equation inf, deterniine all l)ill'ereiitiate both and multiply the former by a multiplier I‘, then 2.l'.'+_7/.+C.+I)a.'=O, ‘_)._I/.'-Ir-;l','+€.'— l)B.=0, 3.1,‘,-= — l’('2a. + Bi) ~ I‘,, 33/' = l’a.'+ BB.) — 1‘.-. Now, substitute these values in the f equation, and l’ becoiiics known; then follow at once all the corrections froin the two last- writteii equations. These corrections being applied to the observed angles, every side in the il‘l:ll)glll:1il0ll has a ( etiiiite value, whieh is obtained by the ordinary method of calculation. A spheroidal triangle differs from a spherical triangle, not only in that the curvatures of the sides are different one from another, but more especially in this that, while in the spherical triangle the normals to the surface at the angular points meet at the centre of the sphere, in the spheroidal triangle the normals at the angles A, B, C meet the axis of revolution of the splicroid in three different points, which we may designate a, ,8, y respectively. Now the angle A of the triangle as ineasured by a tlicodolite is the inclination of the planes l}Ao. and CAo., and the angle at B is that contained by the planes ABB and CBB. But the planes Alla. and ABB containing the line AB in common cut the surface in two distinct plane curves. In order, therefore, that a spheroidal triangle may be exactly defined, it is necessary that the nature of the lines joining the three vertices be stated. In a matlieniatical point of view the most natural definition is that the sides be geodesic or shortest lines. Gauss, in his most elegant treatise entitled .I)i.sr1u.is'itz'om's _r/e.'I.er((les circa s21peI_'/ides curvas, has entered fully into the subject of geodesic triangles, and has iii- vestigated expressions for the angles of a geodesic triangle whose sides are given, not certainly finite expressioiis, but approximations inclusive of small quantities of the fourth order, the side of the triangle or its ratio to the radius of the nearly spherical surface being a small quantity of the first order. The terms of the fourth order, as given by Gauss for any surface in general, are very complicated even when the surface is a spheroid. If we retain small quanti- ties of the second order only, and put "33, LE for the angles of the geodesic triangle, while A, B, U are those of a plane triangle having sides equal respectively to those of the geodesic triangle, then, 0' being the area of the triangle and
 * 1, I1, 1: the measures of curvature at the angular points,

g=A+1%(2'.i+h+t), ga=i3+i‘g(-.:+2i:+:), tt'=C +1—‘f)(-.:+b+2c). The geodesic line being the shortest that can be drawn on any surface between two given points, we may be con- ducted to its most lI11l)01‘t'111t characteristics by the follow- ing considerations: lct p, q be adjacent points on a curved surface; through s the middle point of the chord pq imagine a plane drawn perpendicular to gig, and let S be any point in the intersection of this plane with the surface ; then when sS is a normal to the surface; hence it follows that of all plane curves on the surface joining p, 7, when those points are indefinitely near to one another, that is the shortest which is made by the normal plane. That is to say, the osculatiiig plane at any point of a geodesic line contains the normal to the surface at that point. Imagine now three points in space, A, B, C, such that A 15 = B0 = c; let the direction cosincs of A13 be 1, m,n, those of lit) 1', in’, n’, then .2:,g/,2 being the coordinates of B, those of A and C will be respectively——- a'—cl :3/—cm :z—cn
 * S + S7 is evidently least when sS is a minimuin, which is
 * z:+cl’ :3/+cm’ ::+cn'.

Hence the coordinates of the middle point iI of AC are ac + :.‘,c(l' -1), _1/+ _',r(m' — m), 2+ ,'3c(n' — 7L), and the direction