540 Sir W. R. Hamilton’s Theorem connected with the 
We shall then have 
fm) =4t+7%, S (®o) = figt 725 f (U3) = Ag+ %35f (24) 
— hy+q X49 S (4s) i 7X59 e@ecvceoce (9.) 
and the result (4.) of the elimination of « between the equations 
(1.) and (2.), may be expressed as follows : 
0 = (y-Q’ #,—hy) (y—Q! x.—hg) (y—Q! x3—hs) 
(y—Q’ xy—hy) (Y—Q’! 5). seveeneee (102) 
Comparing (10.) with (4.), and observing that the form of 
the equation (1.) gives the relations 
oO = Ly PLgoALst+ Xyt 255 erecsvece (11.) 
O = By otha Ug t%3 Xyt%yUyt+%5 2, 
HX Pet ly Lyb Myst Hy LAH; Wyy eevee (12.) 
O H 2 XX yHXqg Ug Uy 3 Uy U5 TX Xs Uy +25 2 Lo 
4H, Ug ¥y+ Xo LL +Hy Us XU, +H, H, Let Xe XyXyzy  (138.) 
we easily find these expressions for A’ and C’, namely, 
A! — (hi thgthz+h,) 5 eoecesccccee (14) 
and 
C= — Q? (hv thy xg ths xg + hy ry) 
+Q! hy hg (x +X) +h hg (x +23) + hy hy (ay +24) \ 
thighs (q+ 3) thigh, (&o + 4) thy hg (%3 +24) 
— (Ay highs +h, hghy+hy hg hythg hz hg) severe (15.) 
If, then, the coefficient C’, as well as A’, is to vanish inde- 
pendently of Q, and consequently of Q’, we must have the 
four following equations : 
= hth ths+hy; Peco eee res see ser ees eernes ose (16.) 
0 = har thy re ths ay t+hy tgs cevvesecseeeeee (17) 
0 = hy hy (4, +X_) +h hy (x, +3) +h, hy (2, +4) 
thighs (%q+ a3) +hohg (@o+a4) +h hy (x3+%4)3 (18.) 
0 = Ay hghsthyhghythy hghythghghy3 serve (19.) 
which give, by elimination of ,, 
0 = hy, (wP—a2P) +h (x—aP) +hs(xy—xy), — (20-) 
0 = ha, the xoths x34 (hithgths)? x45 (21.) 
O = hathy) (hg +h) (hy tha) + ceveveccecceceeeseeee (22.) 
Of the three factors of the last of these equations, it is mani- 
festly indifferent which we employ ; since the conclusions which 
can be drawn from the consideration of any one of these three 
factors can also be drawn from the consideration of either of 
the other two, by merely interchanging two of the three roots 
21 X_ Xz, without altering the other of those three roots, or 
