. On Cauchy's modulus surfaces. he locus. To find the slope of the tangent to (1) at the double point(x,,o). dx ox a y dx 13 ^y .d^f f dy\ ^-^f- f ^ dy oj- » _dy ^ -d_r -V dx y - Substituting the coordinates of the double point in this equa-tion after putting for x^x ^jB = 12x - Hx^ 6 X c) f = 4xy - ^?x y d = 4x - 6x. Oxoy we get dx = -4v_j^TlBy^- (12x-6x, )(4x-fix.) 4x - 6x, -X . -X -X , 1 As this value is irr^ependent of the Iscation of ti e print o tanp:en-cy in tbe xz - plane, the tv:o branches of the curve through thedouhle point in the xy - projection intersect at a constant angle. To fi
. On Cauchy's modulus surfaces. he locus. To find the slope of the tangent to (1) at the double point(x,,o). dx ox a y dx 13 ^y .d^f f dy\ ^-^f- f ^ dy oj- » _dy ^ -d_r -V dx y - Substituting the coordinates of the double point in this equa-tion after putting for x^x ^jB = 12x - Hx^ 6 X c) f = 4xy - ^?x y d = 4x - 6x. Oxoy we get dx = -4v_j^TlBy^- (12x-6x, )(4x-fix.) 4x - 6x, -X . -X -X , 1 As this value is irr^ependent of the Iscation of ti e print o tanp:en-cy in tbe xz - plane, the tv:o branches of the curve through thedouhle point in the xy - projection intersect at a constant angle. To find the trigonometric tangent of the angle between the twogeometric tangents at the double point. Tan e = 1. - 1. = VT^ = iVs =-fK, © = 120*^Hence the Theorem: Every tangent-plane cuts the Cauchy surface for W ~ l/^in a curve, whose xy projection is a circular cubic with a realdouble point, such thht the branches of the cubic throu^rh the doublepoint intersect at a constant angle of 120f Figure VIII. 2. F1 f. US * I I i rI fi III. Consider the transforination (1) W = az4b = a z~(-b/a)cz-^d c z-(-d/c W = 0 when z = -b/a W =00 when z = -d/c W = a/c z-Zq, where Zo= -b/a, z^= -d/c. Z —Zcjo The factor a/c involves similitude and rotation. W= kz, k= Kafao that as far as the configuration is concerned, nothing is lost in< generality by restricting the investiga- tion to the transf orination \7 = z-Zp,or W = z- A5Z = U^^ V^, it is found that
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Keywords: ., bookcentury1900, bookdecade1910, booksubjecttheses, bookyear1913
