ACTED ON BY NO EXTERNAL EOECES. 765 



So that ii{f-g){g-h){h-f)=^, and kJ^, 



Q.=*^-^f{Q-(5r+/0M+^M}, 

 a=*5i^{Q-(/t+/)M+A/A}, 



Hence 



+((42>j-4r)M'-(4j'+4/)r)M=A + 8!?rMA='-4r'A')Q. 

 Also from the Eulerian equations, 



i'=t^'^YJf{fg+fh-ghr^Sls 



=f{{g-h)ifg+fh-ghfl>i,}, 

 where 



^, = Q.^-{2f+g+h)-KQ.+{fg+fh)KQ.-\-{f^fg+fh+gh)W ■ 

 -(/X5'+/0+2/5'/O^A+/y^A^ 

 But 



A9-h){fg+fh-ghy{fg+fh)=-Xgh{g-h){fg+fli-gny= 

 ^(g-h){fg+fh-ghy(f+fg+fh+gh)=Xf%g-h){fg+fh-ghy 



=^ifg+fh+gh)T(g-h)=-{fg+fh+ghyi, 



^9-h){fg+fh+ghy(f(g+h)-\-2fgh)=2(ff-fVt^)(fg+fh-ghy 



= -Hf9+AW>(fY-ffn=-ifghif9+ff^-^9m^ 

 ti9-h){fg+fh-gh+ghyfgh=fghX(fg-fh){fg+fIt-ghy=^4fYh%. 



' Hence 



P=(4rM+(2»-4pr)A)Q-(2M-2/-A)% 



and 



(An-M")P=(4rMA+(2'-4j>r)A=)ri'-(4rM^ + (22'-4pr)M'A-42rMA'+4r»A^)Q 



+M'(yM-2rA)». 



MDCCCLXVI. 5 M 



