Binomial Theorem and Logarithms: For the Use of the Midshipmen at the Naval School, Philadelphia |
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Página 30
In ( 14 ) substitute for a , b and m the values a = 2x , b = 3y , m = 5 , and we obtain ( 2x — 3y ) 5— ( 2x ) 5—5 ( 2x ) 4 ( 3y ) +10 ( 2x ) 3 ( 3y ) 3 — 10 ( 2x ) 2 ( 3y ) 3 + 5 ( 2x ) ( 3y ) - ( 3y ) 5 , = 32x5-240x + y + 720x3y2 ...
In ( 14 ) substitute for a , b and m the values a = 2x , b = 3y , m = 5 , and we obtain ( 2x — 3y ) 5— ( 2x ) 5—5 ( 2x ) 4 ( 3y ) +10 ( 2x ) 3 ( 3y ) 3 — 10 ( 2x ) 2 ( 3y ) 3 + 5 ( 2x ) ( 3y ) - ( 3y ) 5 , = 32x5-240x + y + 720x3y2 ...
Página 31
This is effected in the same way as in the case of positive integral exponents , by substitution in formulæ ( 13 ) and ( 15 ) , these formulæ having been proved true for any value of the exponent . Therefore , 1 to expand ( a + b ) —1 ...
This is effected in the same way as in the case of positive integral exponents , by substitution in formulæ ( 13 ) and ( 15 ) , these formulæ having been proved true for any value of the exponent . Therefore , 1 to expand ( a + b ) —1 ...
Página 36
This will be effected by making either p or n negative ; therefore for n substitute —n , and we have ( 1x ) x2 + Making n negative in ( 19 ) , we have ( 1 ± x ) ̄ * = 1 = — = — = x + 23 ( n + 1 ) ( 2n + 1 ) = 22 + 2.3.n3 n - " = 1 = x + ...
This will be effected by making either p or n negative ; therefore for n substitute —n , and we have ( 1x ) x2 + Making n negative in ( 19 ) , we have ( 1 ± x ) ̄ * = 1 = — = — = x + 23 ( n + 1 ) ( 2n + 1 ) = 22 + 2.3.n3 n - " = 1 = x + ...
Página 41
... gives _m ( m — 1 ) am 22Xb mam - 1 2 Nm = az b = mam - 1z + m = = ( mo mam - 1 + 2a whence we derive a second value of z , 2ab 2ma " ± ( m - 1 ) b z = b = mam - 1z b mam 2 In this value substitute the value of b = ± ( N — am ) and ...
... gives _m ( m — 1 ) am 22Xb mam - 1 2 Nm = az b = mam - 1z + m = = ( mo mam - 1 + 2a whence we derive a second value of z , 2ab 2ma " ± ( m - 1 ) b z = b = mam - 1z b mam 2 In this value substitute the value of b = ± ( N — am ) and ...
Página 46
By substitution we shall find Let x = 1 + 1 x ' = 2 + · = 2 + · 3 X = 8 1 x ' = 1 + = x " 1 ' ' 2+ 1 -1 + 1 , then we have !!! then 2+ ( $ ) + + 5 x = in which is between 2 and 3 . Let x ' = 2 + 27 Therefore is an approximate value of x ...
By substitution we shall find Let x = 1 + 1 x ' = 2 + · = 2 + · 3 X = 8 1 x ' = 1 + = x " 1 ' ' 2+ 1 -1 + 1 , then we have !!! then 2+ ( $ ) + + 5 x = in which is between 2 and 3 . Let x ' = 2 + 27 Therefore is an approximate value of x ...
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Binomial Theorem and Logarithms: For the Use of the Midshipmen at the Naval ... William Chauvenet Sin vista previa disponible - 2017 |
Términos y frases comunes
3d root 4th root according adding apply approximate assumed base becomes binomial theorem Briggs calculation called CHAPTER characteristic coefficients common compute construction contain convenient convergent correct corresponding cube root decimal denominator determined difference divided division effected employed equal equation evident evolution example expand exponential equation exponents express extract factor find log find the value formula fraction given gives greater Hence indefinitely integral involution involve known less loga manner method modulus multiply naperian logarithm nearly negative neglected Newton number of terms obtain operation places places of decimals positive preceding principle quantity quotient reciprocal reduced remainder represent result rithms root of a² rule shown significant figure simply square root substitute subtracting succeeding terms third true unity whence
Pasajes populares
Página 50 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Página 49 - The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers.
Página 61 - The fourth term is found by multiplying the second and third terms together and dividing by the first § 14O.
Página 50 - The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor.
Página 19 - Cxz+, etc.=A'+B'x+C'z2 + , etc., must be satisfied for each and every value given to x, then the coefficients of the like powers of x in the two members are equal each to each.
Página 74 - The logarithm of a number in any system is equal to the Naperian logarithm of that number multiplied by the modulus of the system.
Página 49 - Corollary. When the base is less than unity, it follows, from art. 3, that the logarithms of all numbers greater than unity are negative, while those of all numbers less than unity are positive. But when, as is almost always...
Página 55 - ... place, the characteristic being positive when this figure is to the left of the units' place, negative when it is to the right of the units' place, and zero when it is in the units
Página 27 - I have no doubt that he made the difcovery himfelf, without any light from Briggs, and that he thought it was new for all powers in general, as it was indeed for roots and quantities with fractional and irrational exponents.
Página 50 - Bee that to divide one number by another, we subtract the log. of the divisor from the log. of the dividend, and the remainder is the log.
Información bibliográfica
| Título | Binomial Theorem and Logarithms: For the Use of the Midshipmen at the Naval School, Philadelphia |
| Autor | William Chauvenet |
| Editor | Perkins & Purves, 1843 |
| Procedencia del original | Universidad de Harvard |
| Digitalizado | 1 Mar 2007 |
| Largo | 92 páginas |
| Exportar cita | BiBTeX EndNote RefMan |