Elements of analysis as applied to the mechanics of engineering and machinery . s of the estimated co-ordinates, which will pass be-tween the points M^N^O^P^Qdetermined by the observedco-ordinates, in such a mannerthat the sum of the squaresof the deviation of the same from these points will be as Fig. 49. t = 0 1 small as possible on both sides. Art. 37. If, in default of a formula for the constant progressionof a magnitude y, or its dependence upon another magnitude ^, itbe necessary to determine a value of the magnitude y which cor-responds to a given value of ^, bymeans of the values of x
Elements of analysis as applied to the mechanics of engineering and machinery . s of the estimated co-ordinates, which will pass be-tween the points M^N^O^P^Qdetermined by the observedco-ordinates, in such a mannerthat the sum of the squaresof the deviation of the same from these points will be as Fig. 49. t = 0 1 small as possible on both sides. Art. 37. If, in default of a formula for the constant progressionof a magnitude y, or its dependence upon another magnitude ^, itbe necessary to determine a value of the magnitude y which cor-responds to a given value of ^, bymeans of the values of x and y asknown by experience or taken froma table, we must make use of the pro-cess of interpolation^ of which onlythe most important part is to be com-municated here. If the abscissas A jr = x. =: ,a?j, and A 31., = ^.,, Fig. 49, andthe corresponding ordinates 3f^ P^ = Vo^ ^^1 A = 2/i, ^^^^t^ ^^2 A = y-i ^i^eC given, the ordinate 31P = y corre-sponding to a new abscissa A3I = ^, may be expressed by the for-mula y = a -\- i3x -\- yoo^, provided the three points P^,P.^,P^ thus. [Art 3T. ELEMENTS OF ANALYSIS. 61 determined, lie in a nearly straight line, or in a slightly curvedarc. If the initial point of the co-ordinates be transferred from A toJi^, the universality does not suffer, but we obtain simply y = afor j; = 0, and consequently, the constant member a = y^. If now we introduce into the assumed equation, first, x^ and y^^and again, oc^ and y^^ we obtain the two following equations of con-dition: y^ — y^ = I3x^ -f- yx,^ and 2/2 — 2/0 ^ /^^2 i~ 7^2: from which there follows 13 = (^{-(A-y.)^ and ^ _ (V: — !/o) ^2 — (^2 — Vo) ^1 / 9 9 • 1 O ** 1 We have, consequently = y I rC^l — ^o) ^^ — (1/2 — Vo) ^^ , fivi — y,) ^2 — (^2 — !/o) ?^i] If the ordinate y^ lay midway between y^ and 2/2, there would bex^ = 2x^^ and then, more simply, 3 !/o — 4 ^^ + ?/2-\ ^ , /-?/„ — 2 y^ -f 2/2 2/= 2/0 (^^.f^O -+(^^) X If onlj^ two pairs of co-ordinates x^^ 3/^,
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