200 
the sinister member of which, it will be observed, coincides 
with 
y-0 (D) e^e y f 
when in this formula we make ?i=2. 
It is noteworthy that the fractions 
2 n — 1 3 n — 2 4 n — 3 n 2 — n -\- 1 
j • • • 
n ’ n ’ n n 
which occur in <j> (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 
y (y' 1- 1 — «) 
n — 1 
for x, and reducing these results 
§ = (S=TFS'"“‘ {(2»-l)(3»- 
+4 (n — 2) (2 n — 1) y n ~ v 
+(n 2) (« 3)} {g:} 
The relations among the differential coefficients may also be 
exhibited under the following forms, viz. 
S? = rf * r ‘ {2(»+l)y— (2»— 1>} 
§ = »’! r* |3(»+1)(2»+1)/ 
— 2 (2w*— 1) (3w+4) xy 
+(2„-l) (3 m 1) x-} { • 
I think it important also to notice a transformation of the 
differential resolvent for the biquadratic similar to that which 
