Examples of the Processes of the Differential and Integral Calculus |
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Page 9
... of Analysis in the Transactions of the British Association ; and to two papers by Mr Great- heed in the Cambridge Mathematical Journal , Vol . 1 . SECT . 1. Functions of One Variable . ( 1 IL Successive Differentiation.
... of Analysis in the Transactions of the British Association ; and to two papers by Mr Great- heed in the Cambridge Mathematical Journal , Vol . 1 . SECT . 1. Functions of One Variable . ( 1 IL Successive Differentiation.
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D. F. Gregory. SECT . 1. Functions of One Variable . ( 1 ) u = x " ; d'u dx = n ( n − 1 ) ... ...... . ( n − r + 1 ) x ” – ” . - ( 2 ) d'u dx u = ( a + bx ) " ; = n ( n − 1 ) ...... ( n − r + 1 ) b ' ( a + bx ) " ~ " ...
D. F. Gregory. SECT . 1. Functions of One Variable . ( 1 ) u = x " ; d'u dx = n ( n − 1 ) ... ...... . ( n − r + 1 ) x ” – ” . - ( 2 ) d'u dx u = ( a + bx ) " ; = n ( n − 1 ) ...... ( n − r + 1 ) b ' ( a + bx ) " ~ " ...
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... 1 ) • + { 3 ′ − ( " + 1 ) 2 ′ + ( e + 1 ) ′ + 1 • Mémoires de l'Académie , 1777 , p . 108 . 1.2 ( r + 1 ) ~ 1 ' } e ( r − 8 ) • + & c . ] SECT . 2 . Functions of Two or more Variables SUCCESSIVE 21 DIFFERENTIATION .
... 1 ) • + { 3 ′ − ( " + 1 ) 2 ′ + ( e + 1 ) ′ + 1 • Mémoires de l'Académie , 1777 , p . 108 . 1.2 ( r + 1 ) ~ 1 ' } e ( r − 8 ) • + & c . ] SECT . 2 . Functions of Two or more Variables SUCCESSIVE 21 DIFFERENTIATION .
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D. F. Gregory. SECT . 2 . Functions of Two or more Variables . If u be a function of two variables x and y , Ex . ( 1 ) dr + s u dy'dx drs u = dx'dy s = 1 , u = xy " ; r = 1 , du dx mxm - 1y " ; = d2 u dy da = m n xm m - 1 1y " - 1 du dy ...
D. F. Gregory. SECT . 2 . Functions of Two or more Variables . If u be a function of two variables x and y , Ex . ( 1 ) dr + s u dy'dx drs u = dx'dy s = 1 , u = xy " ; r = 1 , du dx mxm - 1y " ; = d2 u dy da = m n xm m - 1 1y " - 1 du dy ...
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... x " } - i = 0 , or Σ { ( 2i – n ) Q ; } = 0 . • This extension of a property of Laplace's Functions was communicated to me by Mr Archibald Smith . CHAPTER III . CHANGE OF THE INDEPENDENT VARIABLE . SECT SUCCESSIVE DIFFERENTIATION . 27.
... x " } - i = 0 , or Σ { ( 2i – n ) Q ; } = 0 . • This extension of a property of Laplace's Functions was communicated to me by Mr Archibald Smith . CHAPTER III . CHANGE OF THE INDEPENDENT VARIABLE . SECT SUCCESSIVE DIFFERENTIATION . 27.
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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