A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page xix
... equal roots , f ' ( x ) has m - 1 roots equal to 280 282 284 ... 284 285 ..... them 287 178. Sturm's Theorem 287 179. Examples in which Sturm's Theorem is applied 291 180. The criteria of the number of impossible roots of an equation ...
... equal roots , f ' ( x ) has m - 1 roots equal to 280 282 284 ... 284 285 ..... them 287 178. Sturm's Theorem 287 179. Examples in which Sturm's Theorem is applied 291 180. The criteria of the number of impossible roots of an equation ...
Page 1
... equal fractions each of them is equal to A1 + A2 + A3 + b1 + b2 + b3 + αι a2 a3 an b1 ' ba ' ba ' bu ...... + an + bn ...... and , if m1 , m2 , M3 , m , be any multipliers , to and to ...... M1 α1 + m2 A2 + M3 A3 + m1 b1 + m2 b2 + m3b3 ...
... equal fractions each of them is equal to A1 + A2 + A3 + b1 + b2 + b3 + αι a2 a3 an b1 ' ba ' ba ' bu ...... + an + bn ...... and , if m1 , m2 , M3 , m , be any multipliers , to and to ...... M1 α1 + m2 A2 + M3 A3 + m1 b1 + m2 b2 + m3b3 ...
Page 2
... equal to zero ; so that b1λ1 + b2 ^ 2 + b3λ3 = 0 , whence by elimination , λι = λο b2c3 - c2b3 bac , — cab C11 + C2 ^ 2 + C3λ3 = 0 ; = 13 b1c2 - c1b2 But thus the ratio only of the multipliers has been determined , and therefore any ...
... equal to zero ; so that b1λ1 + b2 ^ 2 + b3λ3 = 0 , whence by elimination , λι = λο b2c3 - c2b3 bac , — cab C11 + C2 ^ 2 + C3λ3 = 0 ; = 13 b1c2 - c1b2 But thus the ratio only of the multipliers has been determined , and therefore any ...
Page 6
... equal to some mean value of the fractions . Q. E. D. Secondly , let b1 , b2 , bз , ... b2 be negative ; then , as before , 01 is > L , < G b1 a2 is > L , < G b2 an bn is > L , < G ; let these inequalities be severally multiplied by the ...
... equal to some mean value of the fractions . Q. E. D. Secondly , let b1 , b2 , bз , ... b2 be negative ; then , as before , 01 is > L , < G b1 a2 is > L , < G b2 an bn is > L , < G ; let these inequalities be severally multiplied by the ...
Page 8
... equal to each other and to the unit , we say that the latter amount is twice or three times or more times the unit ... equal quantities , one to be divisible into two equal parts , and the other into three parts , equal to each other and ...
... equal to each other and to the unit , we say that the latter amount is twice or three times or more times the unit ... equal quantities , one to be divisible into two equal parts , and the other into three parts , equal to each other and ...
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a₁ algebraical angles b₁ becomes Calculus change of sign changes sign circle coefficients constant cosec curve d2F d2F d²u d²x d²y d³u d³y derived function determine differential equation dr dr dr dy dx dx dx dy dx² dy dx dy dy dy dz dy² dz dx equal equicrescent explicit function expression F(xo F(xo+h factor finite quantity fraction func given Hence homogeneous function increases increments indeterminate form infinite infinitesimal Infinitesimal Calculus infinity involved logarithms loge Maclaurin's maxima and minima maximum or minimum minimum value negative partial derived-functions positive primitive equation radius replaced result right-hand member roots Similarly sin x singular value Sturm's Theorem substituting suppose symbols Theorem tion vanish variables variation versin whence zero