A Treatise on the Differential Calculus |
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Page xi
... circles • 239 161 Conditions for the concavity and convexity of the curve towards the pole and for points of inflection . 240 241 243 - CHAPTER XI . On the Methods of tracing the forms of Curves from their Equations . 162 163 164 ...
... circles • 239 161 Conditions for the concavity and convexity of the curve towards the pole and for points of inflection . 240 241 243 - CHAPTER XI . On the Methods of tracing the forms of Curves from their Equations . 162 163 164 ...
Page 204
... circle so described has the same curvature as the curve at the point P. This circle is called the osculating circle , or the circle of curvature at the point P , p the radius , and C the centre of curvature . The equation ( 1 ) shews ...
... circle so described has the same curvature as the curve at the point P. This circle is called the osculating circle , or the circle of curvature at the point P , p the radius , and C the centre of curvature . The equation ( 1 ) shews ...
Page 205
... circle will then degenerate dx2 into a straight line and coalesce with the tangent . Such will be the case , for instance , at points of inflection , where dy is not infinite . If , at any point of the curve , dx dzy dx2 day dx2 = O and ...
... circle will then degenerate dx2 into a straight line and coalesce with the tangent . Such will be the case , for instance , at points of inflection , where dy is not infinite . If , at any point of the curve , dx dzy dx2 day dx2 = O and ...
Page 209
... circle of curvature touches PT , QS , at P , Q , we have , by the nature of a circle , • 2 limit of 2 = limit of ( 2p - 8 ) . 8 , or , since & vanishes in the limit when compared with p , 22 P = limit of 28 Ex . 1. To find the radius of ...
... circle of curvature touches PT , QS , at P , Q , we have , by the nature of a circle , • 2 limit of 2 = limit of ( 2p - 8 ) . 8 , or , since & vanishes in the limit when compared with p , 22 P = limit of 28 Ex . 1. To find the radius of ...
Page 210
... circle of a curve AB at the point P. Then , x , y , A B being the coordinates of P , the inclination of the tangent PT at P to the axis of x , it is plain that - α - x = p sin - and B - y = p cos But cos = ds dy sin , ds = cos y . dy ...
... circle of a curve AB at the point P. Then , x , y , A B being the coordinates of P , the inclination of the tangent PT at P to the axis of x , it is plain that - α - x = p sin - and B - y = p cos But cos = ds dy sin , ds = cos y . dy ...
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Common terms and phrases
algebraical arbitrary functions asymptote axis Cambridge change sign constant cosec curve d'u dy d²r d²u d²x d²z d2z d2z d³u d³x d³y d³z denote df df df dx DIFFERENTIAL CALCULUS differential equations du du dy du² dv₁ dx dx dx dy dx dx dz dx² dx³ dxdy dy dF dy dx dy dy dy dy dz dy₁ dy₂ dy³ dz dx dz dy dz dz eliminate expression f(y₁ find the Differential formula ƒ Y₁ Hence implicit function increment independent variables indeterminate limit maxima and minima maximum or minimum minimum value multiplying negative partial differential coefficients points of inflection positive quantity putting regard shews Suppose tangent Taylor's Theorem total differential Trinity College University of Cambridge whence y+dy Y₁ Y₂ zero бу бх
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