## A treatise on the integral calculus and its applications1857 |

### From inside the book

Page 73

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**tangent**at that point , we have ( Dif . Cal . Art . 358 ) ds dx √ ( 2 ) therefore , 8 = √ ( 8ax ) + C. The constant will be zero if we measure the arc s from the vertex . 73. Application to the Catenary . The equation to the catenary ... Page 78

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**tangent**at that point ; then dr ( Dif . Cal . Art . 310 ) . Let p be the perpendicular cos = ds " from the pole on the same**tangent**; then thus sin p = 2 , therefore cos 4 = √ ( r2 — p2 ) ; r dr _ √ ( 22 — p3 ) ; - 2 ds ds 2 therefore ... Page 79

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**tangent**to the epicycloid at P is cos - - COS a + b b Ꮎ y - y = - - a + b ( x ′ — x ) , sin - sin 0 b where x and y are the co - ordinates of P , and x and y ' the variable co - ordinates . Hence it will be found that the per ... Page 82

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**tangent**at P ; suppose OY = p , PY = u , YOx = 0 ; then p = x cos 0 + y sin 0 , u = x sin 0 -y cos 0 , dy ds = == therefore dp de dx cot 0 , = cosec 0 ; dx dx x sin + y cose + cos + sin 0 dy = - u , de de d'p du dx d02 de Ꮖ cos Ꮎ ... Page 84

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**tangent**at P ; let ACY = 0 . Then ( Plane Co- ordinate Geometry , Art . 196 ) , CY = a √ ( 1 − e2 sin2 0 ) ; therefore AP + PY : = α - √√ ( 1 − e2 sin3 0 ) do , - the constant to be added to the integral is supposed to be so taken ...### Other editions - View all

### Common terms and phrases

Application c² sin² Cambridge circle cloth co-ordinates constant cos² cos³ Crown 8vo curve cycloid definite integral denote differential coefficient double integral dx dx dx dy dz dy dx dz dx element ellipse equal Eulerian integral example expression Fcap Fellow of St Find the area find the volume formula function Hence indefinitely Integral Calculus integrate with respect intrinsic equation John's College latus rectum length limits M.A. Fellow MACMILLAN & CO.'S numerous obtain ordinate parabola partial fractions perpendicular plane positive preceding article radius vector result revolution revolve round round the axis shew sin² solid suppose surface tangent tion tractory transformed integral Trinity College unity vanishes variables vertex x₁ Y₁ πα аф

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