Examples of the Processes of the Differential and Integral Calculus |
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Page 45
... plane curve . ( 11 ) Eliminate the exponentials from € + € y = - Multiply numerator and denominator by e " , then +1 y = - whence e2 = y + 1 and 2x = log y + 1 ' y - 1 y - 1 and differentiating , dy = 1 − y2 . ( 12 ) Eliminate the ...
... plane curve . ( 11 ) Eliminate the exponentials from € + € y = - Multiply numerator and denominator by e " , then +1 y = - whence e2 = y + 1 and 2x = log y + 1 ' y - 1 y - 1 and differentiating , dy = 1 − y2 . ( 12 ) Eliminate the ...
Page 49
... plane . ( 22 ) Eliminate the arbitrary functions from ≈ = ( ay + bx ) . ( ay - bx ) . Taking the logarithm we have - log x = log ( ay + bx ) + log ↓ ( ay – bx ) , and as the functions are arbitrary their logarithms are also arbitrary ...
... plane . ( 22 ) Eliminate the arbitrary functions from ≈ = ( ay + bx ) . ( ay - bx ) . Taking the logarithm we have - log x = log ( ay + bx ) + log ↓ ( ay – bx ) , and as the functions are arbitrary their logarithms are also arbitrary ...
Page 99
... solution of the problem , To find the height at which a light should be placed so that a small plane surface at a given horizontal distance shall receive the greatest illumination from it . ( 11 ) W = X ( a + x 7-2 MAXIMA AND MINIMA . 99.
... solution of the problem , To find the height at which a light should be placed so that a small plane surface at a given horizontal distance shall receive the greatest illumination from it . ( 11 ) W = X ( a + x 7-2 MAXIMA AND MINIMA . 99.
Page 100
... plane may be the greatest possible . ( 13 ) U = - ( sin mx ) ? ( sin x ) 2 , m being an integer . The values of a derived from m tan≈ = tan mæ , make u a maximum . The values of a derived from sin ma = minimum . 0 , make u a The values ...
... plane may be the greatest possible . ( 13 ) U = - ( sin mx ) ? ( sin x ) 2 , m being an integer . The values of a derived from m tan≈ = tan mæ , make u a maximum . The values of a derived from sin ma = minimum . 0 , make u a The values ...
Page 105
... planes each passing through two angles of the prism ; find the inclination of these planes to the axis of the prism in order that for a given content the total surface may be the least possible . = Let ABC abc ( fig . 6 ) be the base of ...
... planes each passing through two angles of the prism ; find the inclination of these planes to the axis of the prism in order that for a given content the total surface may be the least possible . = Let ABC abc ( fig . 6 ) be the base of ...
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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