396 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



Similarly for the coefficient of (2n + 2q-4r + 1 ) D m '"P n+q _. a . in 



we should get from Art. 12 either the expression 



/ _ , y. (n + q r)l (2q 2r) I (n + m)l (n m + q +p 2r) ! 

 ' r! (q-r)\ (n-r)l (2n + 2q -2r + 1)! (n + m-2r + q-p)l 



x fv ( - i \' ( ( 1-PV (+j> + )!(2n-2-r + (/-p-)n 

 *! (<7-^-*)I (?+l>-2r + a)! (n-m-s)\]' 



or the equivalent expression 



,_ y (g+p)l(2n-2r) I (n + q- r)J_ 

 ' r ! (9 - r) l(n-r)l (2n + 2q - 2r + 1) ! 



x^Xt-lY (q ~ p } ' ( n + m + ^ ! ( n -m + 2q-2r-s)n 

 ' si (q-p- 8 )\ (n + m-2r + s)\ (n-m-s)\_\ ' 



14. Hence referring to Art. 1 we see that 



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



-2r+l)l (q-p}\ 



fv / i \ (7 -P) ! (? +P + *"> ! ( 2n ~ 2r + ( J -P ~ s ) ! 1 

 x S ( - 1) 8 y, - r/ w ./. -^rrr , ~V. cos 



s ' \'1~P ' s ') ' (<? +^ ^ r + 6 ') ' ( n + w - s ) U 

 - l) r (2n + 2q - 



(n + q r)l (2q 2r) I (n + m) ! (n m + q +p 2r) ! 

 r\ (q-r)\ (n-r}\ (2n + 2q-2r+ 1)! (n + m + q-p- 2r)l (q-p)\ 



x S / _ iy -.- -- 



> s i( q - p - s )i( q+p -2r + s)\(n-m-s)!] C P) ' 



where r has all values from to 2q, and s has all values from to q p. 



We also see that another form of this expression is 



Q; cos m\ cos p\ 



(q +p) I (2n 2r) ! (n + q r) ! (n + m) ! (n m + q p 2r)l 

 r\ (q-r)l (n-r)\ (2n + 2q-2r+ 1)! (q-p)\ (n-m)l 



(n + m + q +p 2r) ! 



2r s)\(n-m + s)l 



r^ , 



S( 



-T- ^, 



s\(q-p-s)\ (n-m-2r + s)l (n + m-s)\ J 



