548 Mr. stokes, ON THE CRITICAL VALUES OF 



/(.r) = 5„ + 2fi„ cos (30). 



Let A'„, ji"'„...be the coefficients of sin ^ in the direct expansions of /'(a;), f"{x) ... in 



a 



series of sines. If we were to differentiate (30) once we should have B„ for the coefficient 



a 



oi sin . Now 



a 



nir 2 c_, . jiTTX , , 2 „ , . n-irx 2 /.^^ . . n-rrx , 



. - jfix ) cos dai = /(ir ) sin \- - /(a;) sin dx: ; 



a a a a a a a 



and taking the limits of the integrals, and introducing B„ and A'„, we get 



A=--5„ (3,). 



a 



Hence, the series arising from differentiating (30) once gives the direct expansion of /(a;) in a 

 series of sines. 



The coefficient of sin in the series which would be obtained by differentiating (30) /x times, 



all /mr\i' 

 /J. being odd, would be (- 1) 2 — I B„. By proceeding just as in the last article we obtain 



(-1)^ ^:= (-; B„.-^[-;) |/(o)-(- .)"/»} --„(-;) ^/"(o)-(-i)Y'»} + ... 



+ (--^r\.~\r-'{0)-(-l)\r-\a)\ (32). 



When f (0), fip-), &c., are known, this series enables us to develope f^ia;) in a direct series of 

 sines, the direct developement of f{x) in a series of cosines being given. 



15. If we treat the expression for B^ by integration by parts, just as the expression for A„ was 

 treated, going on till we arrive at the integral which gives A'^, and observe that the very same 

 process is used in deducing the value of A'^ from that of S„ as in expanding the latter according to 

 inverse powers of «, and that the index of n in the coefficient of A'^, is - fi, and that .4„ vanishes when 

 « becomes infinite, we shall see that in order to obtain the direct expansion of /*'(>t) we have only 



got to expand B„ as far as — , (the coefficient of — will vanish,) and subtract from B„ these 



terms of the expansion, and then differentiate (30) n times. 



The expansion of B„, at least to a certain number of terms, will proceed according to even 



powers of - , beginning with — . If we suppose that 

 n n 



B„ = 0,— + O3 — + O5 — + ... when n is odd, 

 rr TV n' 



B„ = E, — + E3 — + Ei — + ... when n is even, 



40* (M* Yt" 



(33). 



