200 



the sinister member of wliiclij it will be observed, coincides 

 with 



when in this formula we make ?2=2. 



It is noteworthy that the fractions 



2n—l 3w— 2 4w— 3 n^—n-\-\ 

 J ..* 



n n n n 



which occur in <^ (D), are in arithmetical progression, the 



common difference being . 



The form given at the foot of page 183 for the fourth dif- 

 ferential coefficients may be simplified and brought into 

 striking symmetric relation with the other forms by the 

 elimination of x. In fact, writing 



n—\ 

 for X, and reducing these results 



+4(«— 2) (2«—l )!/■-• 

 +(„_2)(„_3)} {'^} 



The relations among the differential coefficients may also be 

 exhibited under the following forms, viz. 



^^ = ;.>"-« |3(.^+l)(2n+l)/ 



—2 (2w-l) {Zn\^)xy 



+ (2.-1) (3.-1) .^}{^} 



I think it important also to notice a transformation of the 

 differential resolvent for the biquadratic similar to that which 



