INVERSE METHOD OF DEFINITE INTEGRALS. 343 



The successive differential coefficients with respect to x of the func- 

 tions 0, 6' follow simple and elegant laws, thus 



do dQ' 



= 6^'=°'* cos fa: sin + 0}, ^— = e^'=»*« sin {a; sin + 0}, 



d'Q d^Q' 



d^ = 6"^"°'' cos {ar sin + 20}, ^^= e^'^"** sin {« sin + 20}, 



and generally 



d" d° 0' 



• ^-; = e^'°'^ cos {x sin + «0}, -j— = e^ ">"* sin {x sin + «0} . ■ 



Again, the successive integrals relative to x, follow the same laws, 

 omitting the arbitrary constants of integration, 



/^0 = e^cose cos {a; sin 0-0}, /,©' = e^<=°'» sin {« sin 0-0}, 



//e = e^cose cos 1^ sin - 20}, f,'Q' = e^'^"^* sin {x sin 0-20}, 



fj-Q = e^'^"'^ COS far sin0-w0}, f/O' = 6^<=<«« sin {a;sin0-«0}, 



for it will readily be seen by actual differentiation that 



d" d" 



= ^-;; {e^'="'''cos(xsin0-«0)}, 0' = T-^ i^icose sin (a; sin0- m0)}. 



Again, changing the forms of the proposed fimctions, we get 



= 1 {e-'^ + 6"-'^^}, 0' = -4== {e"'^^ - e"''"^'}, 



whence, expanding and passing from the exponential to trigonometrical 

 functions 



a;* of 



= 1 + a; cos + - — - . cos 20 + , . ^, cos 30 + &c. 

 1.2 1.2.3 



0' = a; sin + — — . sin 20 + , ^ „ sin 30 + &c. 



1.2 1.2.3 



