T A Y 



l:!0 



T A Y 



.r, fora moment, we writi- thi- expansion 



4,; +a 1 .r + a j r t + &c. = A, + A,i + A r rM 

 and if we differentiate both side* with respect to a^, x and 

 11 the other coefficients remaining constant, we have 



&C-) - * = 





which shows that cannot enter any coefficient ore- 

 ceding A,, or 



rfA, 



The first side of this is the same series, whatever letter 

 was made to vary ; the second side is therefore always 

 the same series; whence we collect that <i\ m ^ n : <1<i n 

 does not alter with the value of m, being always the co- 

 efficient of j" in the development of <f>'(a t +a,x+, &c. . It 

 is enough to satisfy this condition for each letter and its 

 preceding one ; that is to say, each co-efficient differen- 

 tiated with respect to any one letter, is to yield the same 

 result as the directly preceding co-efficient differentiated 

 with respect to the directly preceding letter. The follow- 

 ing rules are found sufficient. To pass from any one de- 

 rivative of foi to the next, arrange the letters , A, c, &c., 

 ord,,, a,, r &c., whichever may be used, in order, in 

 term: differentiate with respect to the last letter in each 

 term, and multiply by the letter which comes next to 

 it. And when the last but one immediately precedes the 

 last in the alphabet or other consecutive system, do the 

 same with the last but one, and divide by the exponent of 

 the last letter, as it becomes after the increase which it 

 receives from the process of the preceding letter; but in 

 no case, use any letters but the last or the last but one. 

 For instance, beginning with <jxi, in which is only one 

 letter, we have <f>'a . A, or 



<t>'".b; 



in which are two letters, a and A, consecutive. Operate 

 upon b, and we have ^'a.c; operate on <'<i, and \\ e ha\ e 

 again <ji"a . b, which, with the ft which was in before, is 

 ff>"a . ft*, which we divide by the new exponent of 6, or by 

 2, whence 



A"n 

 D"<n = <f>'a.c+ -^-ft'. 



In forming l> n (^i, we use only c in <f>'a . e, because a does 

 not immediately precede r : and we get (the succession 

 being ", f>, r, <./, g, h, k, &c.) 



and so on. As toon however as the law is established, it 

 is best to form a table of the successive derivatives of the 

 powers of ft by this same law : we then have 



as far as -r^-i A" i 



in which 0'a, <"a, &c. arc to be taken from the function 

 by common differentiation, and the derivatives of the 

 powers of A from the table. This being done, we have 



r, Scc.) = 

 -, ^ 



nnd the process is shortened to its utmost extent ; all that 

 ii not differentiation being merely reference to a table and 

 writing the result. 



\V. shall give materials for proceeding as far : 



. not that so much will often be necessary, 

 (ill because' it i- di-irahle to show with bow little trouble 

 .1' enormous labour KI the ordinary way, Mich, for 

 instance, as that solved in UIAKUM :KI, maybe 



looked at without dismay. We have 1 > form e - 

 live of every power of b, Di", in which m + n doe* not 

 exceed 12. 



bu nfetftaM MUlim. foe (hit of A rbogMl. in which he wlU |mb.Uy In TO 

 rr Mlomrn. Th nd.-ut who U not rmlM by ihU, wd ouuro* proou* 

 Atbojwt k. will Hod Wwft IrwlUn .UutxllDg in ilnintioiii. 



Dft=c T*b=<t 

 A Dft=A 

 D'ft=n D'6=p 

 l)A=2ftc 



=2'.A f - 

 D'ft'=26/ + a-A - 

 D6*=26m +2c/+ i'/; + 'Ifli +^' 



=2Am-2cm + - -fh 



D6'= 



D i ft s =36V+3Ac 

 T) 3 b'=3b i S+6bce+c > 

 Wl>>=3b*g + 66r/+ 36 + 3c 



^aft'A +&>cg+Gbr/+3c 1 S+3f.e > 



D'6'=36'/ +66c/( + 6beh + Mfg+Wfi +6ceg + &/ 



< 



D'6 3 = 3ft*/i + Oic/n -t- 66c/ + Oft/ft + &>gh + 3cV + CceA 



D 3 ft < =46Y+ 126 > c +46^ 



+ Cft/ 4 



D 8 ft 4 =46'' 



l'J^'A + 2-! 

 ce 



(!',- -' 



D6 5 =5ft 4 c 

 Wb i =5b l e+ 106'c* 

 P 3 6 s =5A 4 /+20fr s c-r- lO&V 



4- 10 



V'-c-' 



aM /i.- -i ."' ! ' Vef 







D6=6fte 

 D'ft'=6ftV+156 4 c 



D 4 ft= 6AV +30AV/+ 1 56 4 P +60A*rr+ 1 5A*c 



rA +306V-+ 1 5fty 



' +30ftcV-r-c 



- 



1 ) '/: - s ', 



A/+'> (/ ' >-r7'' 4 .- 



l)'A 9 =9AV4-3CA7c' 



DA'=9Ay+ 



DA'"=10A*c 



