208 
PROFESSOR A. SCHUSTER ON SOME HEFINITE INTEGRALS, 
it is more convenient to take the form of Gauss, for whicli Thomsox and Tait have 
adopted the symbol 0^^, and which is connected with by the relation 
0 
<7 
— O' ) ■ 
{In - 1)L 
Another constant, which has been a])plied ])y Adolf Schmidt, is based on the 
consideration that for numerical work it is inconvenient to deal with functions the 
average values of which differ considerably from each other. This author therefore 
takes a function 7"^©^ as basis of calculation, and determines rl, so that the average 
value of the square of cos cr^ over a sphere of unit radius is ecjual to unity. The 
function defined in tliis way is connected with 0;i and Ql by 
li: = (2n - 1) : ! ^ 
/ 
€^(2n + 1 ) 
(«. + a ): {n — a) 1 
0 ! = 
e^(2n + 1) (n — g-)! 
(n + a ); 
'•C/r 
where is etjual to 1 tor cr = 0 and equal to 2 for all other values of cr. 
If a function is to be expanded in terms of Et we must therefore write 
\/ - 
{2n + 1) {n — cr) ’ 
{n + cr)! 
For numerical work the inti^^duction of IE in place of Qt possesses un(h)uhted 
advantages. Although at first I was reluctant to adopt the additioiial complication 
due to the introduction of a scpiare root and the addition of yet another function to 
those given l^y previous writers, I found that the inconvenience of tabulating values 
differing considerably in magnitude from each other was very great, and I therefore 
felt myself almost com])elled during the course of the investigation to adopt Schmidt's 
function as above defined. 
AY e now substitute G bito the etjuations for /E and fcid write : 
?/ = 
.s = 
TT (n — :V) ;! /2n + i 
2(/;+h);GV Gr 
^ i" - G:: +G 
2 (a A ij': V e 
# 
TT G — 3 ) : / 2li -\- 
2 (a +’2);! V E, 
TT {n —_4)E A /^ 
2 (a + 3):; A er 
A. 
V; 
(« 
+ 
a) : 
!(a- 
- cr); 
* 
(a 
+ cr 
— 
1): 
: (ft - 
- c — 
-1):; 
\ii 
+ cr 
— 
1); 
: (77 - 
- cr — 
-1);: 
(71 
+ 
a)'. 
;(a - 
- a)\ 
t 
{n 
+ cr 
— 
1); 
: {n - 
-a — 
■ IV 
(a 
+ 
a)l 
] (77 - 
- <T]\ 
» 
(a 
+ 
G! 
1 (77 - 
- 
' 
(a 
+ a 
— 
1): 
! (77 - 
- cr — 
■ 1):; 
Vh 
Uh 
My 
S 
rr 
R* 
wliei'e = 1 and = 2 if c > 0. 
By substitution of the values of V,L &c., we ]uay also write more simplv 
