Examples of the Processes of the Differential and Integral Calculus |
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Page 9
... results of the labours of mathe- maticians in this field of research are referrred to various Memoirs of Liouville in the Journal de l'Ecole Polytech- nique , Vol . x111 . , and in Crelle's Journal ; to two papers by Professor Kelland ...
... results of the labours of mathe- maticians in this field of research are referrred to various Memoirs of Liouville in the Journal de l'Ecole Polytech- nique , Vol . x111 . , and in Crelle's Journal ; to two papers by Professor Kelland ...
Page 14
... appears that + a X. d d + a X = € dr ( ( ell X ) . This result , when generalized , is of great importance in the solution of Differential Equations . If the function to be differentiated be ( a + 14 SUCCESSIVE DIFFERENTIATION .
... appears that + a X. d d + a X = € dr ( ( ell X ) . This result , when generalized , is of great importance in the solution of Differential Equations . If the function to be differentiated be ( a + 14 SUCCESSIVE DIFFERENTIATION .
Page 18
... results are useful in the theory of definite in- tegrals . In the following examples the functions are 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 ...
... results are useful in the theory of definite in- tegrals . In the following examples the functions are 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 ...
Page 29
... result is of such extreme complexity , that it happens for- tunately that we have seldom to employ these transformations for high orders of differentials ; and where this is necessary , that the nature of the case usually gives us the ...
... result is of such extreme complexity , that it happens for- tunately that we have seldom to employ these transformations for high orders of differentials ; and where this is necessary , that the nature of the case usually gives us the ...
Page 30
... result is d'x + x − € 3 = 0 . dy ( 4 ) Change the variable in du ገ dy ( 1+ y ) = ( 1 from y to a , when x = log { y + ( 1 + y2 ) * } . du a The result is dx + u = — ( e * + € ̄1 ) . ( 5 ) Change the variable in y2 ď u dy du + Ay + Bu ...
... result is d'x + x − € 3 = 0 . dy ( 4 ) Change the variable in du ገ dy ( 1+ y ) = ( 1 from y to a , when x = log { y + ( 1 + y2 ) * } . du a The result is dx + u = — ( e * + € ̄1 ) . ( 5 ) Change the variable in y2 ď u dy du + Ay + Bu ...
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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