86
Tal
686]
which is
which attending to the before-mentioned values of 77, £ is
= 2 sin %t cos (2n% — £ + 77) t,
and the formula thus is
S S'_
R\ n + S R'\ n + S' ’
(RS' — R'S) V 1 _ i2 sin gt cos (2nÇ — £ + 77) t dt
(R\ n + S) (R'\ n + S') J sinh 7rt
We have here
cos (2n£— £ + 77) t = cos 2n%t cos (77 — £) t — sin 2nty sin (77 — £) t,
whence summing from 1 to go by means of the formulas
cos 2££ + cos
sin 2 %t 4- sin 4$ + ... = ^ cot £i,
(which series however are not convergent), the numerator under the integral sign
becomes
sin %t {— cos (77 — £) £ — cot sin (77 — £) £},
which is
sin %t sin r)t
and the formula thus is
v {IIS' — R'S) \ n _ f sin %t sin rjt dt
(RX n + S) (R'X n + S') J sin £t sinh 7rt 5
and we therefore find
_ A V{(a — d)~ + 4,bc\ (AD — BG) fsin %t sin rjt dt
(pX Q (cl> _ J g j n ip gi n h ^ ’
which is the result in question.
The solution is a particular one; calling it for a moment (cpx), then, if the
general solution be (px = <£>&• + (px), it at once appears that we must have <&x — < $>x 1 = 0 ;
RS'
and as it has been shown that .37^ is a function of x which remains unaltered by
RS
/RS'\
the change of x into x 1} this is satisfied by assuming <5>x = f > an arbitrary
RS'
Hence we may to the foregoing expression of <px add this term
function of
Postscript. The new formula
_ (\ n+1 - 1 )(ax + b) + (\ n - X) (- dx + b)
X ~ (X n+1 — 1) (cx + d) + (X n — X) ( cx — a)’
where
1 a? + d? + 2 be
^ X ad —be
C. X.