A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page 28
... and the function is discontinuous . A is called a point of discontinuity . As an instance of a function of this description the following may be mentioned . Replacing the circular quanti- 28 [ 15 . ON EXPLICIT AND IMPLICIT FUNCTIONS .
... and the function is discontinuous . A is called a point of discontinuity . As an instance of a function of this description the following may be mentioned . Replacing the circular quanti- 28 [ 15 . ON EXPLICIT AND IMPLICIT FUNCTIONS .
Page 29
Bartholomew Price. the following may be mentioned . Replacing the circular quanti- ties by their exponential values , it may easily be proved that sin ( a - 3 ) cos a + cos ( a + ß ) + cos ( a + 2ß ) + ... ad infin . = B 2 sin Suppose ...
Bartholomew Price. the following may be mentioned . Replacing the circular quanti- ties by their exponential values , it may easily be proved that sin ( a - 3 ) cos a + cos ( a + ß ) + cos ( a + 2ß ) + ... ad infin . = B 2 sin Suppose ...
Page 42
... replacing therefore e - 1 by its equivalent in the above equa- tion , and omitting do when added to the finite quantity x , we have If therefore f ( x ) = dy = ex dx { ex + 1 } 2 ° ex " f ' ( x ) = { ex + 1 } 2 ex ex + 1 Ex . 4. To ...
... replacing therefore e - 1 by its equivalent in the above equa- tion , and omitting do when added to the finite quantity x , we have If therefore f ( x ) = dy = ex dx { ex + 1 } 2 ° ex " f ' ( x ) = { ex + 1 } 2 ex ex + 1 Ex . 4. To ...
Page 43
... replacing the sine of an infinitesimal arc by the arc itself , and the cosine by unity , we have dy = ( cosx - da sin x ) ( sin 2x + 2 dx cos 2x ) — cos x sin 2æ , dy = 2 dx cos x cos 2x - da sin x sin 2x , ― omitting , as is necessary ...
... replacing the sine of an infinitesimal arc by the arc itself , and the cosine by unity , we have dy = ( cosx - da sin x ) ( sin 2x + 2 dx cos 2x ) — cos x sin 2æ , dy = 2 dx cos x cos 2x - da sin x sin 2x , ― omitting , as is necessary ...
Page 46
... replaced by the arc to the radius unity , and its cosine by unity . Hence we have Perimeter of circle π 2na- = 2πα ... replacing π tan П by n in accordance with Lemma II , we have Convex surface of cone = nb { a2 + b2 } ± = πb { a2 + b2 } ...
... replaced by the arc to the radius unity , and its cosine by unity . Hence we have Perimeter of circle π 2na- = 2πα ... replacing π tan П by n in accordance with Lemma II , we have Convex surface of cone = nb { a2 + b2 } ± = πb { a2 + b2 } ...
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a₁ algebraical angles b₁ Calculus change of sign changes sign circle coefficients constant curve d2F d2F d²u d²x d²y d2y dx2 d³u d³y derived determine differential equation dr dr dx dx dx dy dx² dy dx dy dy dy dz dy² equal equicrescent explicit function expression F(xo F(xo+h factors finite quantity func geometrical given Hence homogeneous function increases increments indeterminate form infinitesimal Infinitesimal Calculus infinity involved Let f(x logarithm Maclaurin's maxima and minima maximum or minimum minimum value negative partial derived-functions particular values plane positive primitive equation proper fraction real roots replaced result right-hand member Similarly singular value straight line Sturm's Theorem substituting suppose symbols Taylor's Series Theorem tion vanish variables variation versin whence zero
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