100 Mr. Adams on [Feb. 



Example 2. — Let it be required to develope u = . 



u =;J-^=(a + ^)- =«-•+/;.= A +fz /.A=i, • 

 1^:^ -(a + %)-=- a- +/,« =3 B +/;«.•.©= -L, 

 ^ =2(a + *)-»= 2 a-3 +f,,z= 2C+/,;..-.C= 1, 

 |^= -2.3(a+z)-* = -2.3a-*+/,„;i;=2.3D + 



/.n «.••!> = - ;^» 

 &c 



Therefore, by writing the values of A, B, C, &c. in the pro-- 



blem, we have = i+-^"~::i+ &c. 



Example 3. — To find the log. (a + x) in a series, log. of base 

 unity. 



u = l{a + z) = l{l + '-)a=:l(l +^^ + la = la + 



fz = A -^ fz ,\ I a =: A, 



iJL = _i_ = 1 +/, , = B +/, :. .-.1 = B, 



^ = - (fl + z)-^= - «-' +/n ;. = 2C +/„ 2 .-. - 1 



= 2C, 

 ^ = 2(a + z)-'=2a-' + f,,,z = 2.3J) + f,,, z r,\^ 



= 3D, 



&c 



By substituting for A, B, C, &c. we have 



log. (a + =.) = / . « + i - 5^ + 3^ - j^ + &c. 



Example 4. — To find the log. (a — x) in a series, log. of base 

 being unity. 



« =r /(a - X) = l(^[ ^V\a = 1 .a +/2= A + fz.-. 

 ^= - Cl^3 = - (I +/,.,-) = 2 . 3 D +/„. ..-.D 



1 



&c vij^.ci^j^^ 



