Examples of the Processes of the Differential and Integral Calculus |
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Page 16
... suppose . - Developing u { ( 1 + h2 } " . u ' e2 h ) 2 + h " by the binomial 2 и ( 2 u ) 2 theorem , we have u ' u ' e2 u " { ( 1 + h ) 2 " + n ( 1 + h ) 2n − 2 h2 2u 2 и ( 2 u ) * n ( n - 1 ) u ' + ( 1 + -h ) 2n - 4 1.2 2 u ( 2u ) 1 h ...
... suppose . - Developing u { ( 1 + h2 } " . u ' e2 h ) 2 + h " by the binomial 2 и ( 2 u ) 2 theorem , we have u ' u ' e2 u " { ( 1 + h ) 2 " + n ( 1 + h ) 2n − 2 h2 2u 2 и ( 2 u ) * n ( n - 1 ) u ' + ( 1 + -h ) 2n - 4 1.2 2 u ( 2u ) 1 h ...
Page 37
... suppose ≈ to vary while x and y are con- stant , we find dx = R.dr , so that finally dx dy dx MNR dp dq dr . = 2 The general expression for M is complicated , and it is of little use to give it here , as the consideration of the ...
... suppose ≈ to vary while x and y are con- stant , we find dx = R.dr , so that finally dx dy dx MNR dp dq dr . = 2 The general expression for M is complicated , and it is of little use to give it here , as the consideration of the ...
Page 41
... suppose V = 1 , fffdx dy do is the expression for the volume of any solid referred to rectangular co - ordinates : and it becomes fffr2 dr sine de dp when referred to polar co - ordinates . ( 10 ) Having given a function of x and CHANGE ...
... suppose V = 1 , fffdx dy do is the expression for the volume of any solid referred to rectangular co - ordinates : and it becomes fffr2 dr sine de dp when referred to polar co - ordinates . ( 10 ) Having given a function of x and CHANGE ...
Page 49
... suppose . dy Differentiating with respect to x , dx dy d2z dz dz dz dx dx dy Differentiating with respect to y , dz dz dz 2 = f ' ( x ) dx dy dz dz 3 dx dy dy dx dy = ƒ ' ( x ) dz dz Multiplying by dy ' da " and subtracting , dx dz 2 ...
... suppose . dy Differentiating with respect to x , dx dy d2z dz dz dz dx dx dy Differentiating with respect to y , dz dz dz 2 = f ' ( x ) dx dy dz dz 3 dx dy dy dx dy = ƒ ' ( x ) dz dz Multiplying by dy ' da " and subtracting , dx dz 2 ...
Page 94
... SUPPOSE that u is any explicit function of a : the follow- ing rule will enable us to determine those values of a which render u a maximum or minimum . 66 du Equate to zero or dx infinity let a be a possible value of a obtained from ...
... SUPPOSE that u is any explicit function of a : the follow- ing rule will enable us to determine those values of a which render u a maximum or minimum . 66 du Equate to zero or dx infinity let a be a possible value of a obtained from ...
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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