452 ADDITION TO THE MEMOIR ON TSCHIRNHAUSEN’s TRANSFORMATION. [358 
2. It is found by developing that the right-hand side is in fact divisible by 
BD — C 2 , and that the quotient is 
= (- 1 +1 O0 2 - 90 4 ) (B 2 + D 2 ) 2 
+ (80 + 160 s - 240 5 ) (B 2 + D 2 ) BD 
+ (4 + 80 2 + 40 4 -160«) B 2 D 2 
+ (- 640 s -1920 5 ) (B 2 + D 2 ) G 2 
+ (160 2 - 4160 4 -1120«) BDG 2 
+ (- 1280 4 + 1280«) G\ 
3. This is found to be 
= -1 2 U' 2 +12 JU'H' + 4IH' 2 
- 8IJU'%' 
-16J 2 ©' 2 , 
which is consequently the value of — M. We have therefore 
_ O'* = jjj' 3 _ IU' 2 H' + 4 H' 3 
+ (I 2 U' 2 -12 JU'H' - 4IH' 2 ) ©' 
+ 8 IJU'W 2 
+ 16J 2 ®' 3 , 
which is the required identical equation. 
Article No. 4. Calculation of the Cubinvariant. 
4. We have 
= (H - A/®') {/ U' 2 - 3H' 2 + (12 JU' + 2IH') 0' + 7 2 ®' 2 } 
- <t>' 2 , 
whence, substituting for — <f>' 2 its value and reducing, we find 
J* = JU' 3 + 0'. f I 2 U' 2 + ®' 2 (4IJU) + ®' 3 (16/ 2 - & I 3 ). 
Article No. 5. Final expressions of the tiuo Invariants. 
The value of I* has been already mentioned to be I* = IU' 2 + 0T2 JU' + ®' 2 . |/ 2 , 
and it hence appears that the values of the two invariants may be written 
/* = (I, 18/, 3I 2 \U, f®') 2 , 
J* = (/, I\ 9//, — / 3 + 54/ 2 3£t/ 7 , f®') 3 .