Examples of the Processes of the Differential and Integral Calculus |
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Page 20
... . + ( r + & c . 1 ( 27 ) Let u = € + 1 We might in this case expand the function and differen- tiater times each term in the development , but as this would give d'u dx expressed in an infinite series , 20 SUCCESSIVE DIFFERENTIATION .
... . + ( r + & c . 1 ( 27 ) Let u = € + 1 We might in this case expand the function and differen- tiater times each term in the development , but as this would give d'u dx expressed in an infinite series , 20 SUCCESSIVE DIFFERENTIATION .
Page 21
D. F. Gregory. would give d'u dx expressed in an infinite series , the following method , due to Laplace * , is to be preferred . It is easily seen on effecting two or three differentiations that the form d'u of dx must be z a‚é TM 3 + α ...
D. F. Gregory. would give d'u dx expressed in an infinite series , the following method , due to Laplace * , is to be preferred . It is easily seen on effecting two or three differentiations that the form d'u of dx must be z a‚é TM 3 + α ...
Page 69
... of Functions by particular methods . The preceding Theorems sometimes fail from the function which is to be expanded becoming infinite or indeterminate for particular values of the variable , and , more DEVELOPMENT OF FUNCTIONS . 69.
... of Functions by particular methods . The preceding Theorems sometimes fail from the function which is to be expanded becoming infinite or indeterminate for particular values of the variable , and , more DEVELOPMENT OF FUNCTIONS . 69.
Page 87
... infinite , and u , taking the form , is still indetermi- nate unless the factor which becomes infinite should happen to divide out . To determine the value of a function u = 1 becomes for a a , we assume P = Q P1 Q u = = when a = 0 . P ...
... infinite , and u , taking the form , is still indetermi- nate unless the factor which becomes infinite should happen to divide out . To determine the value of a function u = 1 becomes for a a , we assume P = Q P1 Q u = = when a = 0 . P ...
Page 95
... infinite : then , if the first of these dif- ferential coefficients which does not vanish , for the particular value of x , be of an even order , u will be a maximum or a minimum accordingly as this differential coefficient is negative ...
... infinite : then , if the first of these dif- ferential coefficients which does not vanish , for the particular value of x , be of an even order , u will be a maximum or a minimum accordingly as this differential coefficient is negative ...
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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