An Elementary Treatise on Curves, Functions, and Forces
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1852 edition. Excerpt: ...is then reduced to Bt Vl + Ct - + M1 = 0. (307) 194. Scholium. If A, is zero and H, is not so, b and c can satisfy equations (2S6) and (287), and a can be taken to satisfy the equation so that (283) is then reduced to-Bi?! +Ci'l + #i 2 = 0. (308) 195. Scholium. If A1 and J3, are zero, c can be taken to satisfy equation (287), and if either If, or 7, is not zero, a or & can be taken to satisfy the equation M, = 0, so that (283) is then reduced to C, + Ht x2 + 7, y2 = 0. (309) But if both if, and I, are also zero, (283) becomes C, + M1 = 0. (310) 196. Scholium. If the values of 4, B1, C, and Mt have all the same sign, (288) is impossible, and there is no locus. Cases of quadratic locus in space. called the axes of the surface in either of the three last articles, so that the three different values of which are found from equation (304), are the semiaxes. 201. Scholhim. If Ml is zero, the equations (311), (313), and (315) are impossible, but in this case (288) becomes A, x + B, y+ C, z = 0. (317) 202. Scholium. If A1, B1, and C1 have all the same sign, (317) is only satisfied by the values x2 = 0, y2-0, z2 = 0, (318) so that the origin of x2, y2, zs is in this case the required locus., 203. Corollary. If of the three quantities, A1, B, C, one, as C1, is negative, while the other two are positive, we will take A=hB = k-l=k (319) 204. The form of a surface is best investigated byexamining the character of its curved sections, which Examples of quadratic loci. are made by different planes. The farther investigation of the surfaces, represented by quadratic equations, will, therefore, be reserved for Chapter IX. 205. Examples Involving Plane Quadratic Loci. I. To find the locus of all the points in a plane, which are so...
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- 01 Apr 2013
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- Illustrations, black and white