A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page 264
... real roots ; and , when expanded , becomes ( 3 — ( A + B + C ) 02 + ( BC + CA + A B — E2 — F2 — G2 ) 0 - ( ABC + 2EFG - AE2 - BF2 - CG2 ) = 0. ( 39 ) Of this equation the three roots are to be of the same sign , and the result is a ...
... real roots ; and , when expanded , becomes ( 3 — ( A + B + C ) 02 + ( BC + CA + A B — E2 — F2 — G2 ) 0 - ( ABC + 2EFG - AE2 - BF2 - CG2 ) = 0. ( 39 ) Of this equation the three roots are to be of the same sign , and the result is a ...
Page 279
... real or imaginary , and the coefficient of the highest power of a being unity . ... Pn A root of such an expression ... roots and the coefficients , between two equations the roots of one of which are symmetrical functions of those of ...
... real or imaginary , and the coefficient of the highest power of a being unity . ... Pn A root of such an expression ... roots and the coefficients , between two equations the roots of one of which are symmetrical functions of those of ...
Page 280
... root . In other words , I shall prove that a value , real or imaginary , exists , which when substituted for x in f ( x ) makes ƒ ( x ) = 0 . If it is thought that an unfair assumption is made in the following Articles in the extension ...
... root . In other words , I shall prove that a value , real or imaginary , exists , which when substituted for x in f ( x ) makes ƒ ( x ) = 0 . If it is thought that an unfair assumption is made in the following Articles in the extension ...
Page 281
... real root : and that root will be positive or nega- tive according as the constant term of the equation is negative or positive . Hence also it follows that an equation of even dimensions has two real roots if the last term is negative ...
... real root : and that root will be positive or nega- tive according as the constant term of the equation is negative or positive . Hence also it follows that an equation of even dimensions has two real roots if the last term is negative ...
Page 282
... − 1 and of ƒ ( y + z√√ −1 ) , and 0 and T being real circular arcs ; then , if it can be shewn that some value of is such as to render R = 282 [ 173 . ON THE ROOTS OF EQUATIONS . Proof that every equation has a root.
... − 1 and of ƒ ( y + z√√ −1 ) , and 0 and T being real circular arcs ; then , if it can be shewn that some value of is such as to render R = 282 [ 173 . ON THE ROOTS OF EQUATIONS . Proof that every equation has a root.
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Common terms and phrases
a₁ algebraical ascending powers b₁ becomes calculated change of sign changes sign coefficients constant curve d²u d²x d²y d2y dx2 d³u d³y denominator determine differential equation dr dr dr dx dx dr dx dx dx dy dx dz dx² dx2 dy2 dy d2u dy dx dy dy dy dz dy² dy³ dz dx dz dz equal equicrescent Evaluate explicit function expression F(xo F(xo+h factor finite quantity fraction given Hence homogeneous function increases increments indeterminate form infinite infinitesimal Infinitesimal Calculus infinity logarithm loge maxima and minima minimum value negative partial derived-functions particular values positive primitive equation proper fraction replaced result roots Similarly sin x singular value substituting suppose symbols Taylor's Series tion total differentials vanish variation whence x+▲x zero
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