Examples of the Processes of the Differential and Integral Calculus |
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Page 29
... radius of curvature when a is the independent variable is dy + da - d'y dx2 When y is made the independent variable , it becomes { 1+ (税) dy dx dy ( 3 ) Transform d2 y dx2 ' dy 3 CHANGE OF THE INDEPENDENT VARIABLE . 29.
... radius of curvature when a is the independent variable is dy + da - d'y dx2 When y is made the independent variable , it becomes { 1+ (税) dy dx dy ( 3 ) Transform d2 y dx2 ' dy 3 CHANGE OF THE INDEPENDENT VARIABLE . 29.
Page 44
... = -- d y dx dy dx 2 dy dx Substituting these values of yẞ and x - a , we have + - dy 213 x 12 , d2y da 2 in which a and ẞ no longer appear . This is the expression for the square of the radius 44 ELIMINATION OF CONSTANTS AND FUNCTIONS .
... = -- d y dx dy dx 2 dy dx Substituting these values of yẞ and x - a , we have + - dy 213 x 12 , d2y da 2 in which a and ẞ no longer appear . This is the expression for the square of the radius 44 ELIMINATION OF CONSTANTS AND FUNCTIONS .
Page 45
D. F. Gregory. This is the expression for the square of the radius of curvature of any curve . ( 9 ) Eliminate m from the equation ( a + mẞ ) ( x2 - my3 ) = my2 ; the result is axy - dx ' + ( Ba2 — ay3 — y3 ) dy – Bay = 0 . - dx 2 ' dy ...
D. F. Gregory. This is the expression for the square of the radius of curvature of any curve . ( 9 ) Eliminate m from the equation ( a + mẞ ) ( x2 - my3 ) = my2 ; the result is axy - dx ' + ( Ba2 — ay3 — y3 ) dy – Bay = 0 . - dx 2 ' dy ...
Page 103
... radius of the α circle ; then is the half - angle of the segment . - 20 If u be the segment , a u = ax - x2 sin 00 Whence COS If cos a - = 0 , or 00 a maximum . a 818 a 218 ᅲ 1 α COS is a maximum . a a cos x sin 20 a ) = = 0 . the ...
... radius of the α circle ; then is the half - angle of the segment . - 20 If u be the segment , a u = ax - x2 sin 00 Whence COS If cos a - = 0 , or 00 a maximum . a 818 a 218 ᅲ 1 α COS is a maximum . a a cos x sin 20 a ) = = 0 . the ...
Page 104
... radius of the cylinder , y its base . Then y = a determines the cylinder of least surface . a = ( 29 ) The content of a cone being given , find its form when the surface is a maximum . Let πα 3 be the given content , the radius of the ...
... radius of the cylinder , y its base . Then y = a determines the cylinder of least surface . a = ( 29 ) The content of a cone being given , find its form when the surface is a maximum . Let πα 3 be the given content , the radius 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