T A Y 



131 



T A Y 



To verify these results, observe that if we consider each 



letter as of the first dimension, every term of D"b r is o_ 

 the rth dimension ; but if we consider each letter as of the 

 dimension following : 



6 c e f g h k I m n p q 

 1 2 3 4 5 6 7 8 9 10 11 12: 



then every term of DV is of the (+ r)th dimension. To 

 find out if all the proper terms be there, and with the 

 proper exponents, write down the number of ways in 

 which n+r can be made out of r numbers. Thus to 

 verify this point for D'b 3 , write down the ways in which 

 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; 



take the letter answering to each number, in the above 

 list, and multiply the letters of each set together, which 

 gives 



b't, bck, beh, c*h, bfg, ceg, c/ 4 , e*f, 



which are, coefficients exccpted, the terms of D'6 3 in the 

 table. To verify the coefficients separately, observe that 



the coefficient of that term of D"6 r which contains the sth 

 power, ah power, &c., is 



1.2.3.. .. Q-l)r 

 1.2.3. .".*Xl.2.3...<X ..... ' 

 Thus in D 4 A 8 , the term containing Pfe ought to be 

 multiplied by 



1.2.3.4.5.6.7.8 



But the best general mode of verification is derived 

 from the theorem 



dD"b' 



, or 



db 



tliat is, having a certain derivative of a certain power, the 

 lu-xt higher derivative of the next lower power may be 

 1 by differentiating with respect to b, dividing by 

 the exponent of the original power, and then performing 

 the derivation. Thus : 



differentiate with respect to b, and divide by 9. which 



gives 



Now derive, which gives 



86^+5(W/c/+ 286 6 e 2 + 168#eV +704 4 c', 

 the same as is found in the table for D 3+1 6 S ~ l . Here we 

 verify the earlier result of the table from the later : to 

 verify the later from the earlier, use the following: 



n 2 Z /_ ] r 2 



up to c 



1.2 



in which the derivatives of powers of c must be formed 

 from the corresponding tabular ones of b, by changing 

 each letter into the next following. There are thus abun- 

 dant 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 IW is the term GOb-cff; and in 

 passing to DV, we derive from / because it is the last 

 letter ; and from e because, being the last but one, it imme- 

 diately precedes/ in the series. We do not here use b and r. 

 at all ; but if we did use them, we should only repeat tei ms 



ivhich will come into D'6 5 from other sources. Thus: . 



mifn-f gives, ii-om /, 6Qb*ceg, which is set down in ])"// 



'"V/'-r-S, or 306V/ 2 , which is also set down: 



had been used, we should have had (Mb'ivf-^-2, 



i.i- :M>'i-~f, which, on looking, we find set down, as arUmi; 



from the last letter of \ObV. From b, in (Ml/n-f, had it 



been used, v,e should have got 120Acc^/'-4-2, or Utibc-ef, 



'i is also found, and arises from the' last letter of 



. If then we ever find that derivation from one of 



the unused letter^ gives anything but what arises from 



of the letters which are used, it is a sign that some 



has been committed. 



By help of the preceding method, expansions which 

 analysts usually avoid as much as possible, at almost any 

 expense of circumoperation, are carried with the Teatest 

 facility even further than is necessary. The development 

 oi> (a+bx+cx*+ &c.), already given, is one instance; 

 the process in REVERSION OF SERIES is another. This last 

 is done* by expanding .r in powers of ax + bx'' + , &c., by 

 Burmann's Theorem, and making the expansion of the 

 negative powers of (a + bx+cx* + , &c.), which will be 

 wanted, by the method of derivations. We shall state 

 some further applications : 



, &c. 



When m is integer, these derivatives are in the table. 

 When 6 + ex + &c. is a finite series, the whole result is 

 brought out with great ease, compared with the trouble of 

 the common algebraical operation : in this case the value 

 of every letter after the last in the finite series is 0, or the 

 last letter of that series is uot to be employed in derivation. 

 Let the reader try for himself (b+cx+ex^+fx 3 )' by this 

 mode and then in the common way, going only so far in 

 the latter as to feel sure that the former is of no trouble 



compared with it. Let m, m 

 m,, &c. 



m-1 



, &c. be denoted by m, 



(a + bx + ex* + &c.) = a + mba ~ x 



Mt 8 



+ (mac + OTJ&-) a a? 

 + (/wo ! e + OTjaDi* + ; 3 4 3 ) a" "V 



,- m-4 



+ (inaf + w^^rD^o 2 + w? 3 ftD0 3 + m 4 b 4 } ft x 4 



+, &e. ; the law of which is evident, the only thing left 

 being the substitution of the values in the tables instead of 

 the derivatives of b. This form is convenient for fractional 

 or negative powers. The following case is worth exhibit 

 ing separately : 



1 



1 



V-ac 



a + bx+,8tc. 

 - aW + a*e 



* + 



a? 

 u" 



' + a s D"6 8 - a"f 



x* 



, &c. We have avoided the formality of writing Db for 

 c , T>-b for e, &c. 



A + BJ- + Cx* +, &c. _ A^ _ Aft- Ba 

 a + bx + CX L +, &u. ~ a a' 



A (A 8 - ate) - Bai + Ca" 



- Ba(A--ac) 



a* +, &c. 



The law is here evident enough ; the next numerator 

 would be 



A (b' aDi'+r^IW ayj-Ba (b 3 aD6 8 +a 8 e) 

 + Co* (i* - ae) - E 3 i + Fa 4 



The derivatives of the general term b may be readily 

 formed, but the particular cases are more useful ; see the 



derivatives of a in the general form above given. We 

 shall not overload this subject with further examples: 

 enough have been given to show those who require deve- 

 lopments of some extent how much labour they might save. 

 It frequently happens that the form given is not 



(f><;a+ba;+cx*+,&c.')but<j>(a+bx + ^+^3+. &c.J, 



n which case the occurrence of the fractions in the deno- 

 minator renders the process more complicated than it need 



M. Wronski lias civen elegant forms of transformation and development, 

 which are most accessible in Montferrier's ' Dictiunnaire des Sciences Mathe- 

 matiques.' Paris, 18IH. The author of these developments has wrapt himself 

 in a rloud (if (ib-curity. and adopted the tone ofan assailant, with uot a little 

 it tli" manner of a charlatan, which has hindered his really remarkable exten- 

 -ions fioia re, eiviii^' the notice to which they are entitled, and himself from 

 oM.-iinini,' tin- character as a mathematician which no one who reads his works 

 in fir a moment deny him. We do not enter into his methods, because, 

 thi.u-h <>od ill theory, they are not easily used, from their excessive geneiality. 

 Koi instance, in the article on Reversion of Series, in the dictionary cited, the 

 inlhor of which is a partizan of M. Wronski. the results are r ,nie<l as Jar as 

 ur article on that subject, not by the vaunted methods, but by the old 

 .if in, 1, terminate coefficients, an immense labour, after which the nann; 

 .nciertuker is very properly recorded. Torepeat the same processand to 

 carry it two terms further, by Arbogast's and Jiunuaun's methods combine,!, 

 did uot take us thiee hours. 



S2 



