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.