1902.] On Spherical Harmonic Analysis. 99 



When it is therefore desired to retain only terms as far as the nth 

 degree, the Fonrier coefficients need only be calculated as far as 

 p = Ti+l. The position of the earth's magnetic axis, e.g., only 

 depending on the terms of the first degree, is completely determined 

 by the coefficients b. 2 for o- = and a , a 2 for cr = 1, 



3. The symbolical representation of the results of this paper is much 

 facilitated by the introduction of a separate symbol for the product of 

 alternate factors, n.n — 2.n — 4 . . . 1, if n be odd, or n.n-2 

 . . . 2 if n be odd. I propose to write n \ ! for such products, and if a 

 name be required for the product to call it the "alternate factorial" 

 or the " double factorial." Full advantage of the new symbol is only 

 gained by extending its meaning to negative values of n. Its complete 

 definition may then be included in the equations 



n\\ = n(n-2)\\, 1 !! = 1 , 2 !! = 2 . 



From this we may derive when n is negative and odd 



»'! = (-!) 2 (T^p' 



while for n negative and even, the factorial becomes infinitely large. 



4. The calculation of the factors A£ and B^f depends on the values 

 of the definite integrals 



-l +i 

 j cos pOdfx, j Q£ sin p$d[i , 

 -i -i 

 and these may be made to depend on the values of the integrals 

 -fi +i 

 ^Q*sm k $dn and J /xQ^ sin -W/x . 

 -l - -l 



It is proved that 



+i 



QZ smiw/u = c V , — L — v — -L — \ — ^_ it n - cr be even, 



J-f* ' (?i-o-)!! (o--A-2)!! (71. + A.+ 1)!!' 



-l 



= if a - cr be odd, 



+ i 



mQ^ sin A u<hi = c — — * { — ) — l — s ; 1 — — , if n - o- be odd, 



-i 



= if n — o- be even. 



The factor c is equal to 2 or to tt according as <r + A. is even or odd. 



5. The integrals 



+i 



J5' r -1 - 



I 2 



