Examples of the Processes of the Differential and Integral Calculus |
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Page 66
... whole series gives the nth negative power of the least root . ( 9 ) If the equation be cy2 - by + a = 0 , of which the two roots are a , ẞ , then - 1.2 1 1 ne a n ( n { 1 + B b b 5 ) c ( a ( ) n ( n = 4 ) ( n -- 1.2 . 3 b b 3 3 ) c + ...
... whole series gives the nth negative power of the least root . ( 9 ) If the equation be cy2 - by + a = 0 , of which the two roots are a , ẞ , then - 1.2 1 1 ne a n ( n { 1 + B b b 5 ) c ( a ( ) n ( n = 4 ) ( n -- 1.2 . 3 b b 3 3 ) c + ...
Page 105
... whole therefore remains constant . We have then to determine the angle ONS or OSN so that the total surface shall be a minimum . Let AB , the side of the hexagon , AP , the height of the prism , b , OSN = 0 . = = a , Then and QM = a cot ...
... whole therefore remains constant . We have then to determine the angle ONS or OSN so that the total surface shall be a minimum . Let AB , the side of the hexagon , AP , the height of the prism , b , OSN = 0 . = = a , Then and QM = a cot ...
Page 133
... whole area included between the curve , the axis of and any ordinate is equal to twice the triangle formed by the ordinate , the tangent at its extremity and the axis of x ; and the solid formed by the revolution of the curve round ...
... whole area included between the curve , the axis of and any ordinate is equal to twice the triangle formed by the ordinate , the tangent at its extremity and the axis of x ; and the solid formed by the revolution of the curve round ...
Page 136
... whole area of the cycloid is three times that of the generating circle , and this discovery , which was the cause of many disputes between rival claimants to the honour of making it , drew the attention of mathematicians to the study of ...
... whole area of the cycloid is three times that of the generating circle , and this discovery , which was the cause of many disputes between rival claimants to the honour of making it , drew the attention of mathematicians to the study of ...
Page 137
... whole length of the curve is equal to four times the diameter of the generating circle . Pascal discovered the means of finding the area and the centre of gravity of any segment of the curve as well as the content and surface of the ...
... whole length of the curve is equal to four times the diameter of the generating circle . Pascal discovered the means of finding the area and the centre of gravity of any segment of the curve as well as the content and surface of the ...
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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