Examples of the Processes of the Differential and Integral Calculus |
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... Reduction 271 III . Integration of Differential Functions of Two or more Variables .... 282 IV . Integration of Differential Equations .. 291 V. Integration of Differential Equations by Series 340 VI . Partial Differential Equations 351 ...
... Reduction 271 III . Integration of Differential Functions of Two or more Variables .... 282 IV . Integration of Differential Equations .. 291 V. Integration of Differential Equations by Series 340 VI . Partial Differential Equations 351 ...
Page 1
... reduced to the differen- tiation of simpler functions by means of the theorem du = du dy dx dy dx ' y being some function of x , and u some function of y . This theorem may be extended to any number of functions , so that du du dv ds dy ...
... reduced to the differen- tiation of simpler functions by means of the theorem du = du dy dx dy dx ' y being some function of x , and u some function of y . This theorem may be extended to any number of functions , so that du du dv ds dy ...
Page 18
... reduced to the required forms by differentiation in the same way as in Ex . 11 . x du u = ( 22 ) Let ( 1 ) d ( - ) dr v Therefore da ( 1 -x x2 ) } } = dr - 1 dx - 1 ( 1 ( 1 1 and by formula ( B ) , 3.4 ... ( + 1 ) -1 d'u da ( 1 - a2 ) + ...
... reduced to the required forms by differentiation in the same way as in Ex . 11 . x du u = ( 22 ) Let ( 1 ) d ( - ) dr v Therefore da ( 1 -x x2 ) } } = dr - 1 dx - 1 ( 1 ( 1 1 and by formula ( B ) , 3.4 ... ( + 1 ) -1 d'u da ( 1 - a2 ) + ...
Page 50
... reduced to dz f ( a ) + ( a ) 1 = 1 0 . dx 0 ; In the same way , differentiating with respect to y , we have dz $ ( a ) + 4 ( a ) = 0 . dy dz dz dy Since from these two equations it appears that and dx are both functions of a , the one ...
... reduced to dz f ( a ) + ( a ) 1 = 1 0 . dx 0 ; In the same way , differentiating with respect to y , we have dz $ ( a ) + 4 ( a ) = 0 . dy dz dz dy Since from these two equations it appears that and dx are both functions of a , the one ...
Page 72
... reduced to a " when x = m 0 , we have A = α " . Therefore m u = a + ma1a " ( m - · 1 ) ( m2 ) + m 2.3 m m - 1 x + m 3 - - 2 1 a2 + a2a a 1 ) m - 2 x2 aa } m - 323 + & c . a13 + ( m − 1 ) a2 а1 а + аž Euler , Calc . Diff . p . 519 . ( 5 ) ...
... reduced to a " when x = m 0 , we have A = α " . Therefore m u = a + ma1a " ( m - · 1 ) ( m2 ) + m 2.3 m m - 1 x + m 3 - - 2 1 a2 + a2a a 1 ) m - 2 x2 aa } m - 323 + & c . a13 + ( m − 1 ) a2 а1 а + аž Euler , Calc . Diff . p . 519 . ( 5 ) ...
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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