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. 1903 edition. Excerpt: ...(n--3)-way locus together with the 002 of tangent planes, viz. and so on. Each of these different classes of oo"'1 elements satisfying the Pfaflian equation will be denoted by the symbol M, _l; each will form a manifold of united elements with (n-1) ' degrees of freedom.' Thus, when n = 2, that is, in ...
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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. 1903 edition. Excerpt: ...(n--3)-way locus together with the 002 of tangent planes, viz. and so on. Each of these different classes of oo"'1 elements satisfying the Pfaflian equation will be denoted by the symbol M, _l; each will form a manifold of united elements with (n-1) ' degrees of freedom.' Thus, when n = 2, that is, in two-dimensional space, the elements are the points with the straight lines through the points. The symbol Ml will now denote either an infinity of points on some curve together with the corresponding tangents to the curve; or a fixed point with the infinity of straight lines through the point; either of these infinities of elements will satisfy the Pfafiian equation In three-dimensional space there are Q05 elements consisting of points with the planes through them. The symbol M2 will now denote one of three 002 sets of united elements, viz. (1) the points of any surface with the corresponding tangent planes; (2) the infinity of points of any curve together with an infinity of tangent planes passing through each point of this curve; (3) the 002 of planes passing through any fixed point; the elements of any one of these three sets will satisfy the Pfafiian equation 156. We must now consider Lie's definition of an integral of a partial differential equation of the first order; and we need only take the case where the equation is homogeneous, and the dependent variable does not explicitly occur; for any partial differential equation of the first order can be reduced to such a form (F orsyth, Diflerential. Equations, 209). Let f(ml, ..., m, pl, ..., p, l)=0 be such an equation; according to the usual definition [ (ml, ..., m, l) = 0 is said to be an integral if, and only if, as as.. f(ml, ..., m, 3-05;, ...
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