Examples of the Processes of the Differential and Integral Calculus |
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Page 41
... co - ordinates in space . If we 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 ...
... co - ordinates in space . If we 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 ...
Page 102
... co - ordinates of A and B be a , b , a1 , b . Then u = AP + BP = { b2 + ( x − a ) 2 } 1 + { b , 2 + ( a , − x ) 2 } 1 = minimum . Whence x - a = - -x { b2 + ( x − a ) 2 } } ̄ ̄ ̄ { b , 2 + ( a ̧ − x ) 2 } § ' - or the angles APM ...
... co - ordinates of A and B be a , b , a1 , b . Then u = AP + BP = { b2 + ( x − a ) 2 } 1 + { b , 2 + ( a , − x ) 2 } 1 = minimum . Whence x - a = - -x { b2 + ( x − a ) 2 } } ̄ ̄ ̄ { b , 2 + ( a ̧ − x ) 2 } § ' - or the angles APM ...
Page 119
... co - ordinates of the ex- tremities of the least distance ( 3 ) , = ( x − x ' ) 2 + ( y − y ' ) 2 + ( ≈ − x ' ) 2 - - is to be a minimum , the variables being subject to the con- ditions ( 1 ) and ( 2 ) . Differentiating , 0 = ( x ...
... co - ordinates of the ex- tremities of the least distance ( 3 ) , = ( x − x ' ) 2 + ( y − y ' ) 2 + ( ≈ − x ' ) 2 - - is to be a minimum , the variables being subject to the con- ditions ( 1 ) and ( 2 ) . Differentiating , 0 = ( x ...
Page 138
... co - ordinates of a point in this curve are , putting CO = a , CNx , CN = y , COQ = 0 , x = α ( 1 cos 0 ) , y = - αθ . It has points of contrary flexure at the extremities D and d of an ordinate passing through the centre of the gene ...
... co - ordinates of a point in this curve are , putting CO = a , CNx , CN = y , COQ = 0 , x = α ( 1 cos 0 ) , y = - αθ . It has points of contrary flexure at the extremities D and d of an ordinate passing through the centre of the gene ...
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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