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When It Is Divided By The Three Numbers 28, 19, And 15 Respectively The Three Quotients Shall Be 10, 2, And 4.

31But he did not attack the question of the representation of products or quotients of directed lines.

31In virtue of relations (2), the change of entropy of a substance between any two states depends only on the initial and final states, and may be reckoned along any reversible path, not necessarily isothermal, by dividing each small increment of heat, dH, by the temperature, 0, at which it is acquired, and taking the sum or integral of the quotients, dH/o, so obtained.

20In complete partition the successive quotients are 77H 2 and 31___12_1 = o .

20In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the I last partial quotient, a n as a n - I +1.

10+ 1/n+ 271+ ., where, after the n th partial quotient, the cycle of partial quotients b 1, b2, ..., b n recur in the same order, is the type of a recurring simple continued fraction.

10If bl, b2, ..., k J is the cycle of recurring quotients, then b n 2a1, b, bn_1, b 2 bn_2, b 3 = bn_3, &c.

10In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.

10, p, ,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I +bnpn -2 1 qn = anq,i l +bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 + b2= q2.

10Specific gravity bottles are also adjusted at 60° F., the figures indicating specific gravities being quotients obtained by dividing in each instance the weight of the solid or liquid by the weight of an equal bulk of water, both taken at 60 F.

11If r be the number of quotients in the recurring cycle, we can by writing down the relations connectin g the successive p's and q's obtain a linear relation connecting p nr +m, t'(n-1)r +m, +m in which the coefficients are all constants.

11Hence the successive remainders are successively smaller multiples of L, but still integral multiples, so that the series of quotients k, s, t,.

00Let X, Y, And Z Be The Three Quotients Of The Divisions; The Number Sought Will Then Be Expressed By 28 X Io, By 19 Y 2, Or By 15 Z 4.

00are called the successive partial quotients.

003.3 (iii.) Multiples, Submultiples and Quotients

00We first construct the multiple-table C, and then subtract successively zoo times, 30 times and I times; these numbers being the partial quotients.

00Treating x as the unknown quotient corresponding to the original dividend, we obtain successive dividends corresponding to quotients x-200, x- 230 and x-231.

00are called the quotients.

00, obtained by taking account of the successive quotients, are called convergents, i.e.

00Thirdly, the entire vis viva of the system or, as it is now called, the energy, which is obtained by multiplying the mass of each body into half the square of its velocity, is equal to the sum of the quotients formed by dividing the product of every pair of the masses, taken two and two, by their distance apart, with the addition of a constant depending on the original conditions of the system.

00This simple procedure gives the first few quotients of, that we listed above in equation 12.

00Band exposure quotient The sum of the frequency exposure quotients of a band at a location.

00Hence the successive remainders are successively smaller multiples of L, but still integral multiples, so that the series of quotients k, s, t,.

00In virtue of relations (2), the change of entropy of a substance between any two states depends only on the initial and final states, and may be reckoned along any reversible path, not necessarily isothermal, by dividing each small increment of heat, dH, by the temperature, 0, at which it is acquired, and taking the sum or integral of the quotients, dH/o, so obtained.

00Specific gravity bottles are also adjusted at 60Ã‚° F., the figures indicating specific gravities being quotients obtained by dividing in each instance the weight of the solid or liquid by the weight of an equal bulk of water, both taken at 60 F.

00When It Is Divided By The Three Numbers 28, 19, And 15 Respectively The Three Quotients Shall Be 10, 2, And 4.

00Let X, Y, And Z Be The Three Quotients Of The Divisions; The Number Sought Will Then Be Expressed By 28 X Io, By 19 Y 2, Or By 15 Z 4.

00But he did not attack the question of the representation of products or quotients of directed lines.

00are called the successive partial quotients.

00In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the I last partial quotient, a n as a n - I +1.

00+ 1/n+ 271+ ., where, after the n th partial quotient, the cycle of partial quotients b 1, b2, ..., b n recur in the same order, is the type of a recurring simple continued fraction.

00If bl, b2, ..., k J is the cycle of recurring quotients, then b n 2a1, b, bn_1, b 2 bn_2, b 3 = bn_3, &c.

00In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.

00If r be the number of quotients in the recurring cycle, we can by writing down the relations connectin g the successive p's and q's obtain a linear relation connecting p nr +m, t'(n-1)r +m, +m in which the coefficients are all constants.

00, p, ,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I +bnpn -2 1 qn = anq,i l +bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 + b2= q2.

003.3 (iii.) Multiples, Submultiples and Quotients

00We first construct the multiple-table C, and then subtract successively zoo times, 30 times and I times; these numbers being the partial quotients.

00Treating x as the unknown quotient corresponding to the original dividend, we obtain successive dividends corresponding to quotients x-200, x- 230 and x-231.

00In complete partition the successive quotients are 77H 2 and 31___12_1 = o .

00are called the quotients.

00, obtained by taking account of the successive quotients, are called convergents, i.e.

00Thirdly, the entire vis viva of the system or, as it is now called, the energy, which is obtained by multiplying the mass of each body into half the square of its velocity, is equal to the sum of the quotients formed by dividing the product of every pair of the masses, taken two and two, by their distance apart, with the addition of a constant depending on the original conditions of the system.

00This simple procedure gives the first few quotients of, that we listed above in equation 12.

00Band Exposure Quotient The sum of the frequency exposure quotients of a band at a location.

00

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