244 PROCEEDINGS OP THE AMERICAN ACADEMY 



From which 



To find m' 



— ^%sv + vs;.;')) 



= S(g-).fiy — gSl^ilfiy) 



^ g^S).[iy — gSlfi.Sl^y. 

 In the same way 



S(g)LjM.qpj') = g'^Xfiy — gSlft.SXi-iy. 

 But 



S(<jpLg)f*.»') == S(5rl 4- ).''li)(gn + /r^)/ 



= S(yU,. + 2 ^Ay + ^a«)y 



= g^Sluy — X-fi^SXfiy. 

 Hence 



m'=2g'' — 2 gSX^i + /- — Xy = 3 (/^ ^ 2 ^SA/i — P.^^. 



To find m" : — 



^(fhfi.v = S(g). + ?.'70/'-}' = 9^k^r 



SX.cp[i.v = Sl(gfi -\- fi'X)y = gSlfiy. 

 But 



SLft.qpv = S{X.fi.(gy -(- XS/ij' — ySlfi -)- jmSPv/)) 



= S(gXiiy — SXfi.lf^iy) = (g — SX[i)SX[iy. 

 And therefore, 



m" = g -\- g-\- g—SXn = 3 g— SP./i. 

 To find qj'y, 



m = 97 + ^-Sf^J' — J'S V + f^SP.j' = y/ + YXy{i ; 

 hence 



g,2j- = (pcpy = g(gy _|_ VAj'^) + Y {X(gy + YXyfl)fl), 



= ^7 + 2^VA,'^ + V(^V^j7..,.) 



= 5'V + 2 5'(^-S/^?' — ?'SV + f^^k) + /^^'%r 



-}- l|M^SP.^ — XSX^S^y -\- ySV,fi — ^SljwSXy. 



