Examples of the Processes of the Differential and Integral Calculus |
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... Geometrical Problems Involving the Solution of Differential Equations 440 XI . Evaluation of Definite Integrals 464 XII . Comparison of Transcendents 506 DIFFERENTIAL CALCULUS . CHAPTER I. DIFFERENTIATION . Functions of One X CONTENTS .
... Geometrical Problems Involving the Solution of Differential Equations 440 XI . Evaluation of Definite Integrals 464 XII . Comparison of Transcendents 506 DIFFERENTIAL CALCULUS . CHAPTER I. DIFFERENTIATION . Functions of One X CONTENTS .
Page 61
... involves really two different functions of x , which in the develop- ment are given by means of the double sign . ( 11 ) Let u3 - 6ux - When x = = 0 , 8 = 0 . u3 = 8 , u = = ( 8 ) . The possible root of this is 2 , and if we take it ...
... involves really two different functions of x , which in the develop- ment are given by means of the double sign . ( 11 ) Let u3 - 6ux - When x = = 0 , 8 = 0 . u3 = 8 , u = = ( 8 ) . The possible root of this is 2 , and if we take it ...
Page 66
... involving negative powers the result gives us the sum of the nth negative powers of the roots ; while , as has just been stated , the whole series gives the nth negative power of the least root . ( 9 ) If the equation be cy2 - by + a ...
... involving negative powers the result gives us the sum of the nth negative powers of the roots ; while , as has just been stated , the whole series gives the nth negative power of the least root . ( 9 ) If the equation be cy2 - by + a ...
Page 67
... involves positive powers of b a • If in these equations we substitute for - y , and then y find the sum of the inverse nth powers of the roots of the transformed equation , we obtain a series for the direct th powers of the roots of the ...
... involves positive powers of b a • If in these equations we substitute for - y , and then y find the sum of the inverse nth powers of the roots of the transformed equation , we obtain a series for the direct th powers of the roots of the ...
Page 95
... involving dxn - 1 d " u dx all the others vanish when a is put equal to a , so that is reduced to one term . dx " The investigation of the maximum and minimum values of u is sometimes facilitated by the following considerations . If u ...
... involving dxn - 1 d " u dx all the others vanish when a is put equal to a , so that is reduced to one term . dx " The investigation of the maximum and minimum values of u is sometimes facilitated by the following considerations . If u ...
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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