This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1825 Excerpt: ...r.----+v, then, as appears from what has been proved before (p. 118), when r is any whole positive number, a=r, and z=0; that is, z=0 when r is 1, 2, 3, &c. or z contains the factors r-1, r-2, r-3, &c. in inf. (Art. 269); hence, z=Q.r--1..r--2. r--3.&c. in inf., which cannot be expressed in finite terms; consequently, ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1825 Excerpt: ...r.----+v, then, as appears from what has been proved before (p. 118), when r is any whole positive number, a=r, and z=0; that is, z=0 when r is 1, 2, 3, &c. or z contains the factors r-1, r-2, r-3, &c. in inf. (Art. 269); hence, z=Q.r--1..r--2. r--3.&c. in inf., which cannot be expressed in finite terms; consequently, r+z cannot be expressed in finite terms unless Q, that is, unless z=0; and since we know that a, or r + z, may be expressed in finite terms, it follows that z=0, r 1 and that a--r. In the same manner it appears that 6=r.----, &c. and consequently the signs of the terms, in which those odd powers are found, will be changed. 1 2 (234.) Cor. 2. If the index of the power, to which a binomial is to be raised, be a whole positive number, n--1 the series will terminate, because the coefficient n.. '2 n--20.....----.&c. will become nothing, when it is continued to n+1 factors. In all other cases the number of terms will be indefinite. ' (235.) Cor.3. When the index is a whole positive number, the coefficients of the terms taken backward, from the end of the series, are respectively equal to the coefficients of the corresponding terms taken forward, from the beginning. Thus, if a+xbe raised to the 8th power, the coefficients are 1, 8, 28, 56, 70, 56, 28, 8, 1. By subtractingone series from the other, n--1 n--2, _-3 o = wax-1+w..---aV 3 + &c. 2 3 (238.) The trinomial a + b+c may be raised to any power by considering two terms as one factor, and proceeding as before. n--1 Thus, a+b + c" = a'1 + n.b + c.an-'+w.---. b + c a"-z + &c. and the powers of b + c may be determined by the binomial theorem. The theorem by which a multinomial may be raised to any power, is given by Mr. Demoivre, Analyt. page 87. ON SURDS. (239.)...
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