A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page 318
... Hypocycloid , accord- ing as it rolls without or within the fixed circle . We shall consider the epitrochoid to be the normal case , and deduce the equations to the other curves from its equations by changing the signs and values of the ...
... Hypocycloid , accord- ing as it rolls without or within the fixed circle . We shall consider the epitrochoid to be the normal case , and deduce the equations to the other curves from its equations by changing the signs and values of the ...
Page 319
... to polar coordinates , by putting a = a + r cos , y = r sin ; whence r = 2a ( 1 — cos 4 ) . The curve is called the Cardioid , from its heart - like shape . ( 40 ) 206. ] In the equations to the hypocycloid , let 205. ] 319 THE CARDIOID .
... to polar coordinates , by putting a = a + r cos , y = r sin ; whence r = 2a ( 1 — cos 4 ) . The curve is called the Cardioid , from its heart - like shape . ( 40 ) 206. ] In the equations to the hypocycloid , let 205. ] 319 THE CARDIOID .
Page 320
... hypocycloid , let b = a ; whence 2 we have x = a cos y = 0 } ; ( 44 ) which equations express a straight line on the axis of x , of length 2a , which is coincident with the diameter of the circle . SECTION 4. - On certain general ...
... hypocycloid , let b = a ; whence 2 we have x = a cos y = 0 } ; ( 44 ) which equations express a straight line on the axis of x , of length 2a , which is coincident with the diameter of the circle . SECTION 4. - On certain general ...
Page 342
... hypocycloid whose equation is , Art . 206 , x } + y } = a } . The equation to the tangent is હું η + = a ; x & yt and therefore fo = a * x * , and no = = a + y * ; a * y * = a2 ; .. no2 + §o2 = a * ( x * + y3 ) and therefore the ...
... hypocycloid whose equation is , Art . 206 , x } + y } = a } . The equation to the tangent is હું η + = a ; x & yt and therefore fo = a * x * , and no = = a + y * ; a * y * = a2 ; .. no2 + §o2 = a * ( x * + y3 ) and therefore the ...
Page 446
... hypocycloid , whose equation is similarly , Similarly , dy dx = y $ = x + .. xŝ + y = at . day = a s 1+ dx2 3x + y + a * 3xŝ y * y * X Z aš x + dy2 af dx2 = = x + 3x + y + ; n = y + 3x3 y ‡ ; ξτη = = x + 3x * y * + 3x * y * + y , ( x ...
... hypocycloid , whose equation is similarly , Similarly , dy dx = y $ = x + .. xŝ + y = at . day = a s 1+ dx2 3x + y + a * 3xŝ y * y * X Z aš x + dy2 af dx2 = = x + 3x + y + ; n = y + 3x3 y ‡ ; ξτη = = x + 3x * y * + 3x * y * + y , ( x ...
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a₁ algebraical angles b₁ becomes Calculus change of sign changes sign circle coefficients constant cosec curve d2F d2F d²u d²x d²y d³u d³y derived function determine differential equation dr dr dr dy dx dx dx dy dx² dy dx dy dy dy dz dy² dz dx equal equicrescent explicit function expression F(xo F(xo+h factor finite quantity fraction func given Hence homogeneous function increases increments indeterminate form infinite infinitesimal Infinitesimal Calculus infinity involved logarithms loge Maclaurin's maxima and minima maximum or minimum minimum value negative partial derived-functions positive primitive equation radius replaced result right-hand member roots Similarly sin x singular value Sturm's Theorem substituting suppose symbols Theorem tion vanish variables variation versin whence zero