Examples of the Processes of the Differential and Integral Calculus |
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Page 21
... negative indices must disappear of themselves . Hence taking the terms with positive indices only ( ε2 + 1 ) ' + 1 ďu dx ( r + 1 ) = · ( − ) ′ [ 1′e ' * — { 2 ′ · - 1 ' } ( r - 1 ) z ( r + 1 ) ( r + 1 ) r + { 3 ' - 2 " + 1 ' } ( r − 2 ) ...
... negative indices must disappear of themselves . Hence taking the terms with positive indices only ( ε2 + 1 ) ' + 1 ďu dx ( r + 1 ) = · ( − ) ′ [ 1′e ' * — { 2 ′ · - 1 ' } ( r - 1 ) z ( r + 1 ) ( r + 1 ) r + { 3 ' - 2 " + 1 ' } ( r − 2 ) ...
Page 61
... find the 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 .
... find the 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 .
Page 66
... negative powers the result gives us the sum of the nth negative powers of the roots ; while , as has just been stated , the whole series gives the nth negative power of the least root . ( 9 ) If the equation be cy2 - by + a = 0 , of ...
... negative powers the result gives us the sum of the nth negative powers of the roots ; while , as has just been stated , the whole series gives the nth negative power of the least root . ( 9 ) If the equation be cy2 - by + a = 0 , of ...
Page 74
... negative according as ( n - 1 ) r is even or odd . When n is odd the second series terminates ; when n is even it continues to infinity . When n is fractional both series coexist , except for particular values of r . ( 8 ) To expand cos ...
... negative according as ( n - 1 ) r is even or odd . When n is odd the second series terminates ; when n is even it continues to infinity . When n is fractional both series coexist , except for particular values of r . ( 8 ) To expand cos ...
Page 89
... negative , u = 1 ( 35 ) u = 0 1 + x u = 1 - 1 - 2 2 -- - 002 0 = - = = - - ∞ , when X = 1 , , when x = 1 . ( 36 ) u = - u = 20 00 - 1 x log x 1 log x x + 1 ( x − 1 ) log x - = ( 37 ) The sum of the series 8 = ∞ , when a 1 : = , when ...
... negative , u = 1 ( 35 ) u = 0 1 + x u = 1 - 1 - 2 2 -- - 002 0 = - = = - - ∞ , when X = 1 , , when x = 1 . ( 36 ) u = - u = 20 00 - 1 x log x 1 log x x + 1 ( x − 1 ) log x - = ( 37 ) The sum of the series 8 = ∞ , when a 1 : = , when ...
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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