Examples of the Processes of the Differential and Integral Calculus |
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Page iv
... Operation , although the Theory of the process is not usually given in works which are likely to be in the hands of students . I have done so because I think it a matter of some importance that the use of this method should be extended ...
... Operation , although the Theory of the process is not usually given in works which are likely to be in the hands of students . I have done so because I think it a matter of some importance that the use of this method should be extended ...
Page 30
... operation alone , d " 1 d n y " = € dy dx d d d nx T = E to n factors . dx This may be put under the form d { , ( ༦-1 ) , € dx 2 ) a d dx 6 d ( n - 2 ) . -1 dv d dx Now by the theorem given in Ex . 18 , 30 CHANGE OF THE INDEPENDENT ...
... operation alone , d " 1 d n y " = € dy dx d d d nx T = E to n factors . dx This may be put under the form d { , ( ༦-1 ) , € dx 2 ) a d dx 6 d ( n - 2 ) . -1 dv d dx Now by the theorem given in Ex . 18 , 30 CHANGE OF THE INDEPENDENT ...
Page 52
... operation from those of quantity , this theorem may be expressed in a very convenient form , which is useful in various parts of the Integral Calculus : viz . f ( x + h ) = { 1 + h d 12 d dx = € f ( x ) . + + 1.2 da h d3 1.2.3 da3 + & c ...
... operation from those of quantity , this theorem may be expressed in a very convenient form , which is useful in various parts of the Integral Calculus : viz . f ( x + h ) = { 1 + h d 12 d dx = € f ( x ) . + + 1.2 da h d3 1.2.3 da3 + & c ...
Page 79
... operation above described , may be replaced by any equivalent fraction re- sulting from the multiplication or division of both its numerator VI Evaluation of Functions which for certain values of Variable become indeterminate.
... operation above described , may be replaced by any equivalent fraction re- sulting from the multiplication or division of both its numerator VI Evaluation of Functions which for certain values of Variable become indeterminate.
Page 137
... BA ' , beginning from B and if the same operation be performed on AB , beginning from A ' , and so on in succession , the successive involutes will continually approximate to a common cycloid , the GENERATION OF CURVES . 137.
... BA ' , beginning from B and if the same operation be performed on AB , beginning from A ' , and so on in succession , the successive involutes will continually approximate to a common cycloid , the GENERATION OF CURVES . 137.
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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