We shall give materials for proceeding as far as the term not that so much will often be necessary, but because it is desirable to show with how little trouble questions of enormous labour in the ordinary way, such, for instance, as that solved in REVERSION OF SERIES, may be looked at without dismay. We have to form every derivative of every power of 6,D"6 • , in which et+ /1 dots; not ex ceed 12.
he first dimension, every term of n" br is of the rth dimension; but if ve consider each letter as of the dimension following b e efg k l ta npg1 2 3 4 5 6 7 8 9 10 hen every term of D' Is' is of the (it +r)th dimension. To find out f all the proper terms be there, and with the proper exponents, write town the number of ways in which n + r can be made out of r num ,crs. Thus, to verify this point for write down the ways in vhich 10 can be made out of three numbers, namely : 8 + 1 +1, 7+ 2 + 1, 6+3+1, 6+2+2, 5+4+1, 5+3+2, 4+4+2, 4+3+3; ake the letter answering to each number, in the above list, and uultiply the letters of each set together, which gives b'l, Gck, belt, Lfg, ceg, cy,chich are, coefficients excepted, the terms of n'b' in the table. To .crify the coefficients separately, observe that the coefficient of that erru of D' bs which contains the stir power, tth power, &c., is 1 . 2 . 3 (r - 1)r 1 . 2 . 3 ... a x 1.2.3...t x Thus, in WV, the term containing ought to he multiplied by 1.
or 168, as is the case.
2x1' But the best general mode of verification is derived from the ,heorem 1 do.lir fdDllfi' °r " = 1 76-1; ,hat is, having a certain derivative of a certain power, the next higher lerivative of the next lower power may be found by differentiating kith respect to b, dividing by the exponent of the original power, and performing the derivation. Thus : WO= 9bif + 72biee + 8400, lifferentiate with respect to b, and divide by 9, which gives SLY + + 58bles.
Now derive, which gives Sbi g +5Gbnef + + the same as is found in the table for Here we verify the earlier result of the table from the later : to verify the later from the earlier, use the following : r -1 D" = c . r . r n + , &e. .
r(r - 1) (r-n+1) op to 1. 2 in which the derivatives of powers of c must be formed from the corresponding tabular ones of 6, by changing each letter into the next following. There are thus abundant means of verification. We will
mention yet one method more. Only the last letter and the last but one (and that only when the two letters are consecutive) are used in the derivations. if we use any letter, no new term is produced, but only a repetition of those which other terms give. For instance, in is the term and in passing to orb', we derive from f because it is the last letter; and from c because, being the last but one, it immediately precedes f in the series. We do not here use b and c at all ; but if we did use them, we should only repeat terms which will come into bib" from other sources. Thus gives, from f, which is set down in nibs ; from e, or which is also set down ; from c, if c had been used, we should have had 606leef4-2 or which, on looking, we find set down, as arising from the last letter of 1000. From b, in 60blcef, had it been used, we should have got 120bccef÷2, or 60belef, which is also found, and arises from the last letter of If then we ever find that derivation from one of the unused letters gives anything but what arises from some of the letters which are used, it is a sign that some error has been committed.
By help of the preceding method, expansions which analysts usually avoid as much as possible, at almost any expense of eircumoperation, are carried on with the greatest facility even further than is necessary. The development of to (a + bx + &c.), already given, is one instance; the process in Itsvenstosr or SERIES is another. This last is done by expanding x in powers of ax + bxl , &c., by Burmann'e Theorem, and making the expansion of the negative powers of (a+ bx + fie.), which will be wanted, by the method of derivations. We shall state some further applications : (b + &c.)" =6". + rub" . x +Ea" &c.
When at is integer, these derivatives are in the table. IWher 6+ + , &e., is a finite series, the whole result is brought out witl: great ease, compared with the trouble of the common algebraical operation : iu this case, the value of every letter after the last in tin finite series Is 0, or the last letter of that series is not to be employer n derivation. Let the reader try for himself (b+ ex+ fri)s by this mode and then in the common way, going only so far in the latter is to feel sure that the former is of no trouble compared with it. Let 1 m, nx &c., be denoted by m„ &c.