202 
PROFESSOR A. SCHUSTER OX SO.HE DEFIXITE IXTEGRALS, 
By comparing these last equations witli the results obtained when the same 
integrals were expressed in terms of the series and ■'^"e may derive tlie following 
special values ; 
= 0 if or l)e even, p odd, n even and p ^ + 3, 
= 0 ,, cr ,, ,, iJ even, n odd ,, p = n + 3, 
\jI = 0 „ o- „ odd, p „ n „ „ p = n-{- 3, 
= 0 ,, cr ,, ,, p odd, 11 even ,, P^n 3. 
The two series marked and had to be introduced because the series 
for Mb Sh Uh break off as soon as one of the factors becomes zero. But in the 
original summation which gave rise to, e.g., M^, viz. ; 
{n ^ or-l)\\(n-\~2y:.«T+ \ )\\ 
A=i {it' — cr)'! (« + A, + 1)!I (cr — A — 2)!' 
(o- — X — 2)1! begins to he infinite when X = a, and hence the terms of the series will 
drop out until (n — X — 2) is negative, i.e., until X—v. For higher values of X, 
' ” ^ i) veil! be finite again. 
(0--A-2):' ^ 
There is, therefore, a second portion of the series 
not included under Mfi and it is this second portion which aj^pears as Kj!. 
§ 9. Iiclations hetiveen Mj!, S^, Uh Vy 
Certain relations exist betAveen the series Avhich serve to ex 2 :)ress the integrals 
Ql co^ pddp. and Qn sin c/g, and these relations are important for the 
■-1 J-i 
calculation of numerical tables. 
Starting from the equation (F, § 2) 
QH — Ci -2 = (u -f- cr — 2) (jl -}- cr — 3) Qli_; — {il — (T -|- 2) (?l — cr -|- l) QH , 
AAe may multiply by cos pjd or sin pd and integrate on both sides. Expressing the 
r(!sult by means of the ahoA'e series, Ave obtain the folloAving set of equations; 
(„. -f o- - 1) (;i _ .3) M: - {n ~ (t) {n + 2) Xl):_, 
= (?i + a- - 2) {n + 2) MUi; - (n - cr + l) [n - 3) Mr' 
{n + cr) (n - 4) ~ (a - cr - l) {n + 3) S:;_, 
• = (n. + cr - 3) {n + 3) S^Is (u - cr -f 2) [n - 4) Sr" 
(a -r cr ~ 1) [n — 2) Ujl — (a — cr) (a + 1) Ur^ 
= (a + cr - 2) (a + 1) Ur? - (a - cr + 1) {n — 2) Ur" 
(n -f cr) (a — 3) Y;: — (a — cr — 1) (a + 2) Yr 2 
= (a + cr - 3) (a + 2) - (a - cr + 2) (7i - 3) Y,r- 
