148 



that is, 



(1)"... 0= 24.^ . «" + 24. (^^3. «' + -^jT-. -" j. - 



and generally, if we make the difFerential index s = i + 2, and divide by the factorial quantity 

 [*+2]'+*, we shall reproduce the equations (V') [68.] under the form 



where 



fcm=— — • ... X' 



'c...]"^ [«3]'^*" C«'+il"'+' C/sJ^i [^a]''» [a/' 



and «„ « ... /8,, /3a ... are any positive integers which satisfy the following relation : 



< + 2 = 2«, + 3«3 + ." ii + iK+i + ^i +2/3, + ... m. (X'"} 



It is easy to see, that if by these equations we calculate successively the coefficients m('>, wO, as 

 functions of « and w, and if we eliminate u by the assumed relation 



u = z.xv\ (Y"") 



£ being a new variable ; the resulting expressions will be of the form 



