Mr. Jarrett on Algebraic Notation. 97 



whence, by substitution, we get the equation (1). 



Ex. 11. Given u = nt + e .s\n u, to develope u in powers 

 of e. 



By Lagrange's Theorem, if y = z + x.<p {y), 



then /(3/)=/(2) + £g.d-"-^{^^d.-./(^)}. 



Put, therefore, y = u, z = nt, x = e, <p {y) = sin j^, and f{y) =f{u) = u. 

 Then f{z) = z, and d,.f{z) = 1. 



Whence u = nt + S„,,~.d, „,•"-' . sin zT 

 "'|m " ' 



00 y^m-^1 GO 

 12m_l 



(Art. 12.) 



1 C-n,„™ , (-1)"-' 



But sJn^^-' = ^-^, . -S Mam- i . L_i^ . sin 2r- 1 .2, 

 2_ c„,„.„ (-1)-^ _„„. ,_,o™ _L 



and sin zr'^ = -, — -t=. SWam. ) '- — .cos2r2 + |2»i. ■■ - ■ . 



(-4) ^ \m-r S;:- 4 .[m 



Also d,'""^ . sin az = (-1)"-' . a'™-'' . sin az, 



and rf/"-' . cos as = ( - l)" . a"""^ . sin az ; 



.-. 4^"-^■i^i^il^'»-' 

 pip^ . s.-" |2m-i . 1^1^ . 2T3ri^'-:= . (-1)"^' . sin ^7:rr . «, 



J: .Cm lo™_ 1 (~ ^ 



r-i 



, ..„—.S«'r\'im-l.\ — '- . 2r-l|-'"-'.sin 2r-l.z, 



(-4) L_ [m-r 



Fb/. III. P«r< I. N 



