440 



REPORT — 1886. 



¥=-ad-^ 

 + 7bcd 



G= 2^2 

 -2c^d 



^a' — acd 

 -2bH 



+ 36c2 



'-ad? 

 + hcd 



^a'-ad^ 



h' + abd 

 + 2ac- 



b' + 2acd 

 -2bM 



c' + a'^d 

 — 7abc 

 + 6&3 



c' + abd 

 — 4ac2 

 + 3&2c 



¥'; 



¥'■ 



Differentiate (1) andredace. Then 



in which 



(2) 



H= D E'- a 'E--E2 + 3EF ; 

 a 



I=DF'-a'r- ^E2 + 2EG + F2; 

 a 



J=nG'- .'G- — E2 + FG. 

 a 



The elimination of y^ between (1) and (2) gives 



d2B7/"-d%'=(EI-FH)^ + EJ-GH . . (3) 



a result which admits of reduction, since both members contain the dis- 

 criminant as a factor. For, 



EI-FH=n(EF'-E'F) 



_^{E(3cE-36F-aG) + aF2} ; 



EJ-GH= : (EG'-E'G) 



2E, 



-— {E(ciB-3Z;G) + aFG} . 



Now 

 E(3cE-36F-aG) + aF2= : x 



Vc' 



