Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
З цієї книги
Результати 1-5 із 100
Сторінка 13
... Whence , comparing these expressions , it appears that d ( a ) dx ( 15 ) uv = x ^ log ∞ , ď ( uv ) da 12 ( a ) € xx " . · = n ( n − 1 ) ... ( n − r + 1 ) x ” - ” { log x + r . - r ( r− 1 ) 1.2 r ( r− 1 ) ( r− 2 ) 1 ( n − r + 1 ) ...
... Whence , comparing these expressions , it appears that d ( a ) dx ( 15 ) uv = x ^ log ∞ , ď ( uv ) da 12 ( a ) € xx " . · = n ( n − 1 ) ... ( n − r + 1 ) x ” - ” { log x + r . - r ( r− 1 ) 1.2 r ( r− 1 ) ( r− 2 ) 1 ( n − r + 1 ) ...
Сторінка 14
... Whence it appears that d d.x + a a ) X = € az d ax ( 4 ) ( € 112 X ) . This result , when generalized , is of great importance in the solution of Differential Equations . If the function to be differentiated be ( a + 14.
... Whence it appears that d d.x + a a ) X = € az d ax ( 4 ) ( € 112 X ) . This result , when generalized , is of great importance in the solution of Differential Equations . If the function to be differentiated be ( a + 14.
Сторінка 38
... Whence dy ' = dr2 dr αφ αφ ď3p _ x2 + y2 аф ( 2 dedy - dr + + - + " ) . + dx2 and therefore = ďo dre + dr 1 do - = 0 . r dr This equation occurs in researches on the motion of fluids . ф ( 4 ) If dp dp ďp dx2 dy dz2 + + = 0 , we find ...
... Whence dy ' = dr2 dr αφ αφ ď3p _ x2 + y2 аф ( 2 dedy - dr + + - + " ) . + dx2 and therefore = ďo dre + dr 1 do - = 0 . r dr This equation occurs in researches on the motion of fluids . ф ( 4 ) If dp dp ďp dx2 dy dz2 + + = 0 , we find ...
Сторінка 44
... whence y - B " d2y dx2 - 1 + d2y dx2 ( dy = 0 , 2 dx dy d2y da dx Substituting these values of yẞ and a -a , we have 3 + ( d ) } * 2 = dy in which a and ẞ no longer appear . 12 , This is the expression for the square of the radius dx2 ...
... whence y - B " d2y dx2 - 1 + d2y dx2 ( dy = 0 , 2 dx dy d2y da dx Substituting these values of yẞ and a -a , we have 3 + ( d ) } * 2 = dy in which a and ẞ no longer appear . 12 , This is the expression for the square of the radius dx2 ...
Сторінка 45
... denominator by e , then y = y + 1 whence € 2 " = y and differentiating , - dy dx € 2 + 1 1 9 = - and 2x = 1 - y2 . y + 1 log y 1 ( 12 ) Eliminate the power from the equation y ELIMINATION OF CONSTANTS AND FUNCTIONS . 45.
... denominator by e , then y = y + 1 whence € 2 " = y and differentiating , - dy dx € 2 + 1 1 9 = - and 2x = 1 - y2 . y + 1 log y 1 ( 12 ) Eliminate the power from the equation y ELIMINATION OF CONSTANTS AND FUNCTIONS . 45.
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a² b2 a²x² angle arbitrary constant asymptote becomes C₁ c²x² Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy 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 plane of reference radius SECT singular solution spiral Substituting subtangent surface tangent plane theorem triangle vanish whence x²)³