THE SUMS OF PERIODIC SERIES. , 567 



and its first /i - 1 derivatives are supposed to be functions of the same nature as before, whicli 

 are considered between the limits x = and x = os ; and it is moreover supposed that /(.r) 

 decreases as a; increases to os , sufficiently fast to allow ff{a;)dx to be essentially convero-ent at the 

 limit 05 , or else that f{x) vanishes when .^=00, and after a finite value of x never changes from 

 increasing to decreasing nor from decreasing to increasing. 0(/3) or -v/,(/3) denotes the coefficient 



of sin /3.r or cos ^x in the developement of /(«) in a definite integral of the form C 0/3 sin jixdx 



"'■ / ylyi.(i)cos (ixdx, ^^{(i) or >//^(/3) denotes the coefficient of sin /3.r or cos /3a; in the deve- 

 loperaent of the /j"* derivative of f{x). The formulae are 



+ - (— ) {/'(O) - (- !)"/'(«)} - ... (^ Odd) (A), 



(-lyj: = ^-j A--[—) i/(0)-(-l)Y(a)} +...(m even) (B), 



^-'y <= (v) ^"^~aW) 5/(0)-(-ir/'(«)|-... (^odd) (C), 



(-1)'b:= {^y B„ + ~-i~-y "|/(0)-(-l)"/'(«)}-... (m even) (D), 



except when w = 0, in which case we have always 



>?s = ^ {/""'(«) -r"'(o)}, 



Bn being the constant terra in the expansion of f^ (;r) in a series of cosines. 



In the formulae {A), (B), (C), (D) we must stop when we have written the term containing 

 the power 1 or 0, (as the case may be,) of — . The formulae for integrals are 



( - O'^^^^O) -/3".^(/3) - -fi^-'fio) + - (i^-Y'io) - - (m odd) (a), 



IT TT 



(-l)^'^.(/3) = /3''<^(/3) --^''-7(0)+-/3''-7"(0)-... (m even) (6), 



TT 'TT 



( - ^)'^<PAft) = ft'i^di) + -fi'-'no) - ~[i^'Y"(o) + ... (m odd) (o), 



TT TT 



( - l)'^K(/3) =/3->|'(/3) + - /3''-y(0) -- /S-V'CO) + ... (m even) (d), 



"TT TT 



where we must stop with the last term involving a positive power of /3 or the j)owcr zero. 



44. As a first example of the application of the principles contained in Sections I. and II. 

 suppose that we have to determine the value of d) for values of x lying between and «, o and h 

 respectively, from the equation 



-J- + ^ = (75), 



dx' dy" ^ '' 



4 I) 2 



