PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



295 



— (e'"cosa— e^'v/— 1 sina— 1)" (cos ax — V — lsinax)} 

 Let e™ cos a — 1=P cos 6, e" sin a = P sin 6 ; 



then 



ps^e^m _2e'"cos(i + l, and tan0 = 



e^cos a— 1 



gOs.': 



.-. A^e"'"' smax=—y=^ P" {(cos «2X'7r+0 + V— lsin«2A7r+fl) (cosar + V'— 1 sin a «) 

 — (coa2\''jr+d—V—l sin2X''n-+0) (cosax—V—1 sinaa^)} 



gM X pn 



= {V — 1 cos «(X'— X)'7r + sinw(A' — X)7r} 8in(a.i- + ra 6 + « \ + X' ■tt) 



V — 1 



= e""^P" (cosrwTT — V — 1 sin ?■ w tt) sin (a ij? + » + « X tt) 



»• and X being any integers. 



Similar expressions may be obtained for the wth differences of cos a x and 



of e™ ^ cos a x. 



25. We shall now proceed to the demonstration of certain theorems analogous 

 to those in the ordinary calculus of differences. 



Prop. 1. 



1 1 nb n(n-V).b- 



where 'o^=a^■bx or apj^=6. 



By formula (6) ; putting — for 



' (>»-!) 



= — +w A — + 



+ &c. 



Now 



J_ _ _ _AJJ^ _ _ 6 2 Jl - 1 • 2 ■ 6- 



&c, &c. 



1 1 nb n(n-l)b- . 

 = H ^^ &c. 



Prop. 2. A similar result may be obtained from formula (8). 



For 



, ^^ 



... 1 



1.2.6^ 



-&C. 



1 1 nb n(n + r)l^ 

 = H ^ &C. 



Prop. 3. a" m » =«? a^m -i-wac a"~^m ^, + &c. 



For {(1 + a) (1 + A') - l}u^ v^ being an operation on % -Vx may be repeated accord- 

 ing to any law, consequently 



a" w^ v^ = { (1 + a) (1 + a') - 1 }" u^ v^ : and every step in the demonstration is the 

 same as when « is a whole number. ^'^ 



