Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
З цієї книги
Результати 1-5 із 30
Сторінка 74
... vanish . find ( 7 ) To expand sin na in ascending powers of sin a . Proceeding in the same manner as in the last example , we 1.2 n2 sin n x = sin nrπ { 1 - ( sin x ) 2 + - + cos ( n − 1 ) r π { n sin x - n2 ( n2 - 22 ) 1.2.3.4 n ( n2 ...
... vanish . find ( 7 ) To expand sin na in ascending powers of sin a . Proceeding in the same manner as in the last example , we 1.2 n2 sin n x = sin nrπ { 1 - ( sin x ) 2 + - + cos ( n − 1 ) r π { n sin x - n2 ( n2 - 22 ) 1.2.3.4 n ( n2 ...
Сторінка 79
... vanish also make P ' = 0 , and Q ' = 0 , we must differentiate again , and so on in succession , as long as the numerator and the denominator both vanish when a is put equal to a . There- fore we may say generally that the true value of ...
... vanish also make P ' = 0 , and Q ' = 0 , we must differentiate again , and so on in succession , as long as the numerator and the denominator both vanish when a is put equal to a . There- fore we may say generally that the true value of ...
Сторінка 95
... vanish , for the particular value of a , be of an even order , u will be a maximum or a minimum accordingly as this differential coefficient is negative or positive . " If du dx = † ( x ) .v , † ( x ) being an es- sentially positive ...
... vanish , for the particular value of a , be of an even order , u will be a maximum or a minimum accordingly as this differential coefficient is negative or positive . " If du dx = † ( x ) .v , † ( x ) being an es- sentially positive ...
Сторінка 109
... vanish , there will be no maxi- mum or minimum unless the third differentials also vanish , while the values of y ' deduced from the biquadratic d'u d'u d1 u +4 · y ' + 6 . 12 y + 4 dx1 da3 dy dx2 dy d1u dady3 d'u 3 · y ' ' = 0 , dy1 ...
... vanish , there will be no maxi- mum or minimum unless the third differentials also vanish , while the values of y ' deduced from the biquadratic d'u d'u d1 u +4 · y ' + 6 . 12 y + 4 dx1 da3 dy dx2 dy d1u dady3 d'u 3 · y ' ' = 0 , dy1 ...
Сторінка 111
... involving da1 , dx2 , ... da , vanish , that is to say , if we determine them by the conditions M1 = 0 , M2 = 0 , ... M1 = 0 , + Mécanique Analytique , Vol . 1. p . 74 . the variations dæ , dx2 , ... da , are MAXIMA AND MINIMA . 111.
... involving da1 , dx2 , ... da , vanish , that is to say , if we determine them by the conditions M1 = 0 , M2 = 0 , ... M1 = 0 , + Mécanique Analytique , Vol . 1. p . 74 . the variations dæ , dx2 , ... da , are MAXIMA AND MINIMA . 111.
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a² b2 a²x² angle arbitrary constant asymptote becomes C₁ c²x² Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy 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 plane of reference radius SECT singular solution spiral Substituting subtangent surface tangent plane theorem triangle vanish whence x²)³