WITH APPLICATION TO SPHERICAL HARHONIC ANALYSIS. 
187 
A little care is necessary in the interpretation of the square bracket on the right- 
hand side. All the factors of the product contained in it may he positive or 
negative, but when A is intermediate between n and cr, some may be positive and some 
negative. In the latter case, one of the factors will be zero when cr — A is even, and 
in that case the integral on the right-hand side is zero. The above expression does 
not include the special case n = cr. By extending the notation of double factorials 
to negative numbers as defined in § 1, we may also write, including the cases when 
n — A, or both n — A and cr — A, are negative, or when n = cr. 
r+i 
-1 
siiB 9 dfx 
{n + <r - + \)\\{n - X - 2)\\ 
(n — O')!! (cr — X — 2)!! (n -f- X -f 1)!! ^ 
(T he even, 
= 0 if w — cr be odd. 
The integral sin^ ^ c//x reduces to the one just determined with the help of 
• -1 
equation (A). We thus find : 
r+l 
slid 9 dij. — c -— 
I {n 
(». + (7)!! (cr + X)! ! (n - X - 3)!! 
- <7 - 1)!! (cr - X - 2)!; + X + 2)!! 
if n —■ cr be odd, 
= 0 if n — cr be even. 
The factor c takes, as before, the value 2 or rr according as cr fi- A is even or odd. 
For the special case cr = 0, the tesseral harmonics reduce to zonal harmonics, and 
the last equation becomes 
r+i 
j sin^ 9 dfj. = c 
?«!! X 1! (??. — X — 3) 
1 ! 
(« - 1)!! (- X - 2)!! (?i + X -f- 2)!! 
if n be odd. 
If A be even and n > A -j- 2, the fraction on the right-hand side is zero, because in 
that case (— A — 2) ! ! = oc , and the mnnerator remains finite, but if n < A fi- 2, the 
value of remains finite whether A he even or odd, and, in that case, the 
right-hand side may also be written (avoiding negative arguments of the factorials): 
n-l 
i-)-c 
!! X !! X !! 
(n - 1)!! (« + X -f 2)!! (X - w + 1)!! 
If n be odd and > A -j- 2, we may transform the negative factorial and write the 
value of the integral 
A+l 
77 
w!!X!!X!!(7i~X-3)!! 
Oo -1)1! (n + X -I- 2)! 
Similarly we obtain 
2 B 2 
