182 
PROFESSOE A. SCHUSTER OX SOME DEFINITE INTEGRALS. 
the product of a tesseral function and cos p9 or sin the tesseral functions being all 
referred to the same axis. I shall denote, as usual, the factorial product of the 
]iumbers up to n by n!, but have found it necessary to introduce a separate notation 
for the products of successive even or successive odd numbers. I consequently define 
n\ \ = n . {n — 2)! !. 
Starting ivith 1 ! ! = 1, 2 1 ! = 2 , 
it follows that, for positive values of n, 
n ! ! = 71. — 2 . » 4 . . . . 
where the last factor is either 1 or 2, according as n is odd or even. 
AVe may extend the definition to negative values of the argument, for the successive 
substitution of 7i = 1, n = — 1, n = — 3, into the first equation, gives 
(-1)!!=!, (-S)!!^-!, (-5)::=.:’, 
and general!V if n is negative and odd. 
For 7( = 2, the original equation gives 0 I ; = 1, 
for 71 = 0, (~ 2) ! ! = 2 c , 
and similarly for all negative and even values of 77, n ! ! is infinite. The ratio of two 
of these factorials of negative numbers is, however, finite, for it is easilv shewn that 
if m and u be two neoatn'e numbers, whether even or odd. 
One of the advantages of a separate notation for what may be called the " alternate 
or “double” factorial, is due to the fact that it often saves the inconvenience of 
different expressions for odd and even numbers. 
§ 2. Foyniida; of Tran!<formation. 
It is convenient to collect together some equations ivhich will often be reiptired. 
.Vfost of these equations ivill ho found already in previous ivritings, such as IIeixe’s 
i’reatise or AdajMs’ llesearches in Terrestrial Magnetism.” 
As re gards zonal harmonics, it is only necessary to quote the well-known relations : 
