388 ON THE PEODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



Hence the coefficient of /i-"*-* in the product D m P n x D'P q 



1.3.5 ...(27t-l)l .3.5 ... (2q- 



(n-m)\(q-p)l 



Similarly the coefficient of /A '-'"+*-* in D m+p P n+<1 



1.3.5 ...(2w + 2g-l) 



(n + q m p) ! 



Hence the coefficient of D m+p P n+Q in the value of D~P n x D'P, 



1.3. 5 ...(2^-1)1.3.5 ...(2g-l) (n + q- m -p) \ 

 1.3. 5 ... (2n + 2q-l) (n-i)l(g-.p)I' 



Also the coefficient of /t-"""* in (^ 2 - l)*Z>'"P n x Z>*P,, 



1.3.5 ... (2n-l) 1.3._5 ... (2q-l) 

 (n m) ! (g p) \ 



but the coefficient of the same power of p, in D m ~ p P n+ri 



1.3.5...(2w + 2g-l) 



jo _ 2 _ , - - 



(n + q m +p) \ 



therefore the coefficient of D m ~P n+Q in the value of (/r - 1 }"D m P n x D'P, 



1 .3.5... (2w-l) 1 . 3. 5 ...(2g-l) (n + r/-m+_p)! 

 1.3.5 ...(2w + 2(/-l)(n-m)! (g-^))! 



supposing m to be not less than p. 



NOTE. In the value of 2D m P n . D"P q given in Art. 6 above (when 

 q=p + 2) the quantity contained in brackets [ ] may also be expressed 

 in either of the following equivalent forms : 



[2q (2q-i)(n-m-2r+l)(n- m - 2r + 2) 



- 2 (2q - 1) (n - m - 2r + 2) 2r (2n - 2r + 1 ) 

 + 2r (2r - 1) (2n - 2r + 1) (2n - 2r + 2)], 



or \_(n-m-2r + 2)(n-m-2r + l)(2q-2r) (2q-2r-l) 



-2(n-m-2r + 2) (2q - 2r) 2r (n + m) 



or [(n - m - 2r + 2) (n - m - 2r + 1 ) (n + m + 2q - 2r) (n + m + 2q-2r-l) 



-2(n-m-2r + 2)(n + m + 2q-2r-l)(n + 

 + (n + m)(n + m l)(n m+ 1) (n m + 2)']. 



