Examples of the Processes of the Differential and Integral Calculus |
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Page 19
... by 1.2 ... r , we find d'u = dx + € ¤¤2 { c ' ( 2x ) ' + r ( r − 1 ) c ' − 1 ( 2 x ) ' −2 r ( r 1 ) ... ( r− 3 ) 1.2 - c - 2 ( 2x ) ' ' + & c . } ( 26 ) From this we can determine the successive 2-2 SUCCESSIVE DIFFERENTIATION . 19.
... by 1.2 ... r , we find d'u = dx + € ¤¤2 { c ' ( 2x ) ' + r ( r − 1 ) c ' − 1 ( 2 x ) ' −2 r ( r 1 ) ... ( r− 3 ) 1.2 - c - 2 ( 2x ) ' ' + & c . } ( 26 ) From this we can determine the successive 2-2 SUCCESSIVE DIFFERENTIATION . 19.
Page 20
D. F. Gregory. ( 26 ) From this we can determine the successive dif- ferentials of cos r2 and sin a2 . Let u = cos x2 + ( − ) sin a2 Then differentiating by the preceding formula d'u dx = c ( - ) * ‚ 2 { ( − ) 3 ( 2x ) ' + ...
D. F. Gregory. ( 26 ) From this we can determine the successive dif- ferentials of cos r2 and sin a2 . Let u = cos x2 + ( − ) sin a2 Then differentiating by the preceding formula d'u dx = c ( - ) * ‚ 2 { ( − ) 3 ( 2x ) ' + ...
Page 37
... determine da by supposing dy = 0 , and dx = 0 , and then eliminating two of the three quantities dp , dq , dr . Supposing we eliminate the last two we have da Mdp , M being a function of p , q , r . From this it follows that when da = 0 ...
... determine da by supposing dy = 0 , and dx = 0 , and then eliminating two of the three quantities dp , dq , dr . Supposing we eliminate the last two we have da Mdp , M being a function of p , q , r . From this it follows that when da = 0 ...
Page 42
D. F. Gregory. ( 10 ) Having given a function of x and y determined . by the equation y2 + b2 = 1 , it is required to transform ( 1/4 ) } * [ idx dy { 1 + ( da ) 2 + ( - ) } * ffdx into a function of and dx when x = a sin cos p , and ...
D. F. Gregory. ( 10 ) Having given a function of x and y determined . by the equation y2 + b2 = 1 , it is required to transform ( 1/4 ) } * [ idx dy { 1 + ( da ) 2 + ( - ) } * ffdx into a function of and dx when x = a sin cos p , and ...
Page 70
... determining the successive differential coefficients . Recourse must then be had to particular ar- tifices depending on the nature of the function which is given . One of the most useful methods is to assume a series with indeterminate ...
... determining the successive differential coefficients . Recourse must then be had to particular ar- tifices depending on the nature of the function which is given . One of the most useful methods is to assume a series with indeterminate ...
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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