Examples of the Processes of the Differential and Integral Calculus |
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Page 21
... positive indices , the terms in the product of ( 2 ) and ( 3 ) which contain negative indices must disappear of themselves . Hence taking the terms with positive indices only ( ε2 + 1 ) ' + 1 ďu dx ( r + 1 ) = · ( − ) ′ [ 1′e ' * — { 2 ...
... positive indices , the terms in the product of ( 2 ) and ( 3 ) which contain negative indices must disappear of themselves . Hence taking the terms with positive indices only ( ε2 + 1 ) ' + 1 ďu dx ( r + 1 ) = · ( − ) ′ [ 1′e ' * — { 2 ...
Page 61
... series x3 25 u = - - & c . a2 a3 a1 Taking the positive value of a , u = a - x x2 2 3x3 + " & c . 8 a 16a2 Taking the negative value of a , v u = a + + x2 5.23 + & c . 8 a 8a2 ( 13 ) If sin y = v sin ( DEVELOPMENT OF FUNCTIONS . 61.
... series x3 25 u = - - & c . a2 a3 a1 Taking the positive value of a , u = a - x x2 2 3x3 + " & c . 8 a 16a2 Taking the negative value of a , v u = a + + x2 5.23 + & c . 8 a 8a2 ( 13 ) If sin y = v sin ( DEVELOPMENT OF FUNCTIONS . 61.
Page 62
... positive integer . dy Differentiating , cos y = dx sin ( a + y ) + x cos ( a + y ) putting = 0 , y = r , we have r ƒ ' ( 0 ) = Differentiating again , dy COS y dx - sin ( a + r ) sin y COS Iπ = sin a . 2 dy = 2 cos ( a + y ) dx 2 ' dy ...
... positive integer . dy Differentiating , cos y = dx sin ( a + y ) + x cos ( a + y ) putting = 0 , y = r , we have r ƒ ' ( 0 ) = Differentiating again , dy COS y dx - sin ( a + r ) sin y COS Iπ = sin a . 2 dy = 2 cos ( a + y ) dx 2 ' dy ...
Page 66
... 3 3 ) c + & c . } , ( the series only continuing so long as there are positive powers b of that is , negative powers of a a or . b Equations Numériques , p . 225 . ( 10 ) Let a - by + cy = 66 DEVELOPMENT OF FUNCTIONS .
... 3 3 ) c + & c . } , ( the series only continuing so long as there are positive powers b of that is , negative powers of a a or . b Equations Numériques , p . 225 . ( 10 ) Let a - by + cy = 66 DEVELOPMENT OF FUNCTIONS .
Page 67
... positive powers of b a • If in these equations we substitute for - y , and then y find the sum of the inverse nth powers of the roots of the transformed equation , we obtain a series for the direct th powers of the roots of the original ...
... positive powers of b a • If in these equations we substitute for - y , and then y find the sum of the inverse nth powers of the roots of the transformed equation , we obtain a series for the direct th powers of the roots of the original ...
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a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz dz dz eliminate ellipse equal Euler factor formula fraction function Geometry gives Hence hypocycloid infinite intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Multiply negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle University of Cambridge vanish whence x²)³