1821.] a Method of applying Maclaurin*s Theorem, 99 



u = A + (B;j-f- C z^-^-J) z' ^'Ez' + V z' + ^c.) = A+/z 

 4^ = B + (2C;j + 3D2"-+4Es^ + 5Fz*+ 8cc.) = B + 



^ = 2C + (2.3Dz + 3.4E2'= + 4.6F2' + &c.) = : 



2C±f,,z 

 ^ = 2.3D + (2.3.4Ez + 3.4.5Fz^ + &c.) = 2.3 



D +/„ z . r^ 



^ = 2.3.4E+(2.3.4.5F;r+3.4.5.6G;2^4- &c.) 



= 2 .3 .4E + 92 



|^ = 2.3.4.5F + (2.3.4.5.6G;r+ &c.) . . . . = 2 . 



3 . 4 . 5 F + ^i «. 

 &c &c. 



Where ^2,^1 2,^11 z,f^^^z, (p z,^^ 2, &c. denote different func- 

 tions of;?:. 



Example 1 . — Let it be required to develope u — {a -{- z)~. 

 u = (a + :?;)'" = «"* +/;s = A +/2 .-. a'" = A. ;| 



-^ = mCa + x)'"-' = ma''-' -^f,z=B + f,z .-. wza"*-* = B, 



^= w(m~l)(a + 2;)"'-'= tn (??z-l) a"*-" +/„ x: = 2 C + 



/„s:.-.w2(wi-l)a'"-^ = 2C, 

 &c 



Then by writing the value of A, B, C, Sec. in the problem, 



we have (« + 2)'" = a"* + m a"*"' ;« + ^ (^ ^ J "^ ;22 



m(m-l)(m-2)an.-3 ^^ 



+ 1.2.3 » + ^C. 



* The usual method of finding the values of A, B, C, &c. is to suppose the variable 

 quantity in the given equations equal to nothing, by which means other equations are 

 obtained which determine A, B, C, &c. in terms of the given function; thus in the 

 equation u = (a + z)"^ — A + B s + C s' + D ~3 + &c. Suppose x = 0, thea 

 «"» = A, 



-— ■ = m (fl + s)'»-» = B + 2Cs -f SDs*^ -f &c.' Suppose2=0, thenw a"'~» = B, 



a z 



d^ u 



—— -= m im—1) {a + sy-^ = 2 C + 2 . 3 D .:; + &c. Suppose s=0, then m (;« -1> 



d z^ 



a«-2 =2 C the same as before. This latter method is evidently as simple in its 



application as the preceding, but it does not appear to me to be so evident, particularly 



to beginners. If be written for z in the equation ?t=A + B5;+C;:^+Ds5 + &&. 



, , , . . . ^ „ d 11 d^ u Cfi u ^ .... 



we should have ^t = A, a given quantity, therefore — , ■— ^, -— ^, &c. will be respec- 



tiyelj equal to nothing. 



g2 



