386 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



and the other expression is 



1.2.3.4.1.2.1.2.3^4 

 1.2.3.1.2. 1. 1 



Next let r = 3. 



Then s 0, 1, 2 give zero results; also when s = 3 the series is reduced 

 to the single term 



1.2.3.4.5.6 



1.2.3 





1.2.3.4.5.6.1.2.3 , . , 



and the other expression is - = -120, which agrees. 



l.^.o. l.Z.o 



Hence there can be no doubt of the accuracy of this result, which 

 is very curious. 



9. We may obtain the first and last terms in the value of 



D"'P n x D p P, t 

 in a more convenient form. 



The first and last terms in the value of D' l P n in D m +*P will be 



Hence in D m P n x D'P p the first and last terms are 



.. ('2n-l) ,j m+f p 



LS J , 



1.3.5 ...2 



n+p 



nd f 1 x,l-3.5...(2p-l)1.3.5...(2n-2p-l) 



1.8.5...(2+1) 



Multiplying by (2p+l)p we get the value of D m P n x D'P P+1 



.1.3.5...(2p+l)1.8.5...(2n-l)(> t -m+l) 

 1.3.5...(2n+2p-l)(2n+2jp + l) 



