OP THE FORM d m P n /dp m x (P>P q /d^. 383 



From the series obtained for D m P n x D"P p+l by means of two of the 

 above equations and by putting p + 1 for p, we may get the value of 

 the coefficient of D m+p P n+p _ ir ^ in the expression for pD m P n x D p+l P p+2 . 



We can also get the coefficient of D m+p P n+p _ !a . +1 in the expression for 

 D m P n xD p+1 P p+l , and hence we may obtain the value of 2D m P n xD p P p+ , 

 in the form of a series. 



Our object is to find the law of formation of the coefficients of the 

 successive terms. 



Following the same process of successive substitution we obtain in the 

 same manner, when q p + 2, 



2x/>P x/)"P -*{ IV (q+p)l(2n-2r)\(n + g-r)l 



Lj rl(q-r)\(n-r)\(2n + 2q-2r+l)\ 



x \_(2n - 2r + 1) (2n - 2r + 2) (2q - 2r) (2q -2r-l) 



-2(2n-2r+ 1) (2q-2r) (2q-l) (n + m) + (2ql) 2q(n + m) (n + m -I)]. 



The quantity in square brackets may be expressed in the form 



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

 _(2n-2r)l (2q-2r-2)l ~ (2n-2r)\ (2q-2r-l)\ (2q-2)\ (n + m- 1)1 



2ql (n + niyi 



+ ~ 



'-2)! (n + m-2)l_\ ' 

 Here the law of formation is clear. 



7. Applying the same process to the equation 



o T)m p v T)p p _ /9~ i c\ T)m p v T)p p / , o\ rim p ... T)p p 



OJ~f -L n X J-J J. +3 \ P ' / r*-*-S -*- n * -^ *- +-2 \ _/ ' / n P + i ' 



it appears that, when q =p + 3, we have the corresponding coefficient in 

 3! nmp . n,P _</_iV _ (?_+^l ! _(2n-2_r)I (w + g-r)! 



' t 1 ! (o r)I (w ?") ! (2n + 2(7 2T + 1) ! 



\I / \ /\ i / 



(2^-2r)! (2n-2r + 2)! (2q-2r)\ (2q-2)l 



; 3. 



_(2q-2r-3)l (2n-2r)l (2q-2r-2)l (2n-2r)\ (2gr-3)l 



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



(2^-2r-l)! (2n-2r)! (2^-3)! (w + w-2)l ~ 



