Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Результати 1-5 із 61
Сторінка 20
... preceding formula d'u dx = c ( - ) * ‚ 2 { ( − ) 3 ( 2x ) ' + ( − ) ' = ' r ( r − 1 ) ( 2a ) * - * -- - - + ( − ) ' = ' ' ? ' ( ' − 1 ) ... ( r ' − 3 ) 2 - 1.2 Now generally ( - ) = c ( − ) * » } , and ( - ) * ( - ) = COS ( x2 + p ...
... preceding formula d'u dx = c ( - ) * ‚ 2 { ( − ) 3 ( 2x ) ' + ( − ) ' = ' r ( r − 1 ) ( 2a ) * - * -- - - + ( − ) ' = ' ' ? ' ( ' − 1 ) ... ( r ' − 3 ) 2 - 1.2 Now generally ( - ) = c ( − ) * » } , and ( - ) * ( - ) = COS ( x2 + p ...
Сторінка 40
... preceding example d2 V dy + de V d - de V 1 d2 V d p + 1 dv + p2 dq2 ραρ - In exactly the same way , the equations of condition being similar , we find dev d2 V + = d'V 1 + d2 V + d p2 do dx2 dr r2 1 dv - r dr Also , as in the first ...
... preceding example d2 V dy + de V d - de V 1 d2 V d p + 1 dv + p2 dq2 ραρ - In exactly the same way , the equations of condition being similar , we find dev d2 V + = d'V 1 + d2 V + d p2 do dx2 dr r2 1 dv - r dr Also , as in the first ...
Сторінка 55
... preceding example let h = -x , then tan - 1 ( x + h ) = tan - 10 = 0 ; 002 therefore tan - 1 x = sin y . sin y . x + ( sin y ) 2 sin 2y 2 003 + ( sin y ) 3 sin 3 y + ( sin y ) 1 sin 4y + & c . 3 4 π cos y Now tan - 1 x = y , and a coty ...
... preceding example let h = -x , then tan - 1 ( x + h ) = tan - 10 = 0 ; 002 therefore tan - 1 x = sin y . sin y . x + ( sin y ) 2 sin 2y 2 003 + ( sin y ) 3 sin 3 y + ( sin y ) 1 sin 4y + & c . 3 4 π cos y Now tan - 1 x = y , and a coty ...
Сторінка 63
... preceding , made by assuming the given equation in y to be d y = F { ≈ + xp ( y ) } . Then if u = f ( y ) , and if we put ƒ F ( x ) = f1 ( * ) , and __ƒ F ( x ) = ƒ { ' ( x ) , and pF ( x ) = p1 ( ~ ) , dz d u = f ( y ) = ƒ¡ ( * ) + ...
... preceding , made by assuming the given equation in y to be d y = F { ≈ + xp ( y ) } . Then if u = f ( y ) , and if we put ƒ F ( x ) = f1 ( * ) , and __ƒ F ( x ) = ƒ { ' ( x ) , and pF ( x ) = p1 ( ~ ) , dz d u = f ( y ) = ƒ¡ ( * ) + ...
Сторінка 64
... = 1 + a + 2 log a " + 32 ( log a ) 2 a3b or y = " + 2 ° ( log « ) " : 32 a ) 2 See Ex . 15 of the preceding Section . c2 C3 + & c . 1.2 1.2.3 = 1 + ca " , a2 c2 a3c3 + & c . 1. 2 1.2.3 ( 4 ) Let y = a + x log 64 DEVELOPMENT OF FUNCTIONS .
... = 1 + a + 2 log a " + 32 ( log a ) 2 a3b or y = " + 2 ° ( log « ) " : 32 a ) 2 See Ex . 15 of the preceding Section . c2 C3 + & c . 1.2 1.2.3 = 1 + ca " , a2 c2 a3c3 + & c . 1. 2 1.2.3 ( 4 ) Let y = a + x log 64 DEVELOPMENT OF FUNCTIONS .
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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