Examples of the Processes of the Differential and Integral CalculusJ and J. J. Deighton, 1846 - 529 стор. |
З цієї книги
Результати 1-5 із 49
Сторінка 4
... roots and powers , it is generally most convenient to take the differential of the logarithm , or , as it is usually called , the logarithmic differential of the function . ( 31 ) Let u = ( a + x ) TM ( b + x ) " , ( 32 ) ( 33 ) log u ...
... roots and powers , it is generally most convenient to take the differential of the logarithm , or , as it is usually called , the logarithmic differential of the function . ( 31 ) Let u = ( a + x ) TM ( b + x ) " , ( 32 ) ( 33 ) log u ...
Сторінка 61
... root of this is 2 , and if we take it , we find by the same method as in the last example the series 1 003 u = 2 + x - + x1 + & c . 21 2.3 1 2 3.4 • • The other series for u would be found by taking the im- possible values of the cube root ...
... root of this is 2 , and if we take it , we find by the same method as in the last example the series 1 003 u = 2 + x - + x1 + & c . 21 2.3 1 2 3.4 • • The other series for u would be found by taking the im- possible values of the cube root ...
Сторінка 65
... root of the equation . ( 6 ) Let y3 - ay + b = 0 : expand y " in terms of - . a Here ƒ ( x ) = x ” , b $ ( x ) = x3 , ≈ = a Whence bn b2 1 y " = = = { 1 + n n ( n + 5 ) b1 1 n ( n + 7 ) ( n + 8 ) , 1 + a2 a 1.2 a1 a2 1.2.3 + a6 a3 · as ...
... root of the equation . ( 6 ) Let y3 - ay + b = 0 : expand y " in terms of - . a Here ƒ ( x ) = x ” , b $ ( x ) = x3 , ≈ = a Whence bn b2 1 y " = = = { 1 + n n ( n + 5 ) b1 1 n ( n + 7 ) ( n + 8 ) , 1 + a2 a 1.2 a1 a2 1.2.3 + a6 a3 · as ...
Сторінка 66
... root . ( 9 ) If the equation be - cy2 — by + a = of which the two roots are a , ẞ , then 1 + = a 4 - nc a b b + € 0 , n ( n - 3 ) c2 1.2 b2 3 b3 ( ~ ) 2 + & c . } , n ( n − 4 ) ( n − 5 ) c3 1.2.3 - · 2 the series only continuing so ...
... root . ( 9 ) If the equation be - cy2 — by + a = of which the two roots are a , ẞ , then 1 + = a 4 - nc a b b + € 0 , n ( n - 3 ) c2 1.2 b2 3 b3 ( ~ ) 2 + & c . } , n ( n − 4 ) ( n − 5 ) c3 1.2.3 - · 2 the series only continuing so ...
Сторінка 67
... roots of the transformed equation , we obtain a series for the direct th powers of the roots of the original equation . ( 11 ) If we thus transform the equation in Ex . 10 it becomes c - by + ay2 = 0 ; and if a , ẞ be the same ...
... roots of the transformed equation , we obtain a series for the direct th powers of the roots of the original equation . ( 11 ) If we thus transform the equation in Ex . 10 it becomes c - by + ay2 = 0 ; and if a , ẞ be the same ...
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a² b2 a²x² angle arbitrary constant asymptote axis becomes C₁ c²x² Cambridge circle co-ordinates condition curvature curve cycloid cylinder determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz eliminate ellipse equal Euler find the value formula function Geometry gives Hence hypocycloid infinite Integrating with respect intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral maximum minimum Multiply negative origin parabola perpendicular radius radius of curvature singular solution spiral Substituting subtangent surface tangent plane theorem tractory triangle vanish whence x²)³