542 Sir W. R. Hamilton’s Theorem connected with the 
observe that, in general, the result of elimination of any vari- 
able 2 between any two equations of the forms 
O=p tq xt x®t+s' a? 
ace fhe? $rhet . \ (36.) 
is 
Oo= g 73! f qr? —2p! r pl’? +p y! gi? r+! s’ plig! x 
—p! sl! gr ate q' 2 pl! rll? ¢/ r! pl’ gq''r!—2 q' 3 gl 2” +g! sg! q/? 
peg? aly!!! sf yt! 9g! tof Age, ncemanvscecwvcccesces  (37.) 
Applying this general formula to the elimination of x, between 
the equations (35.) and (33.), and making, for that purpose, 
p = 234+2e8, ¢ = —2,4 7 = 0, si 1, 38 
p’ = xf— xata—B, YJ =a,—a, =], ake? 
we find, after some easy reductions, 
O = 42,°—42,)5 a+a,* (8 &—68)+2,? (—8a2 +140 8) 
+a,° (6 at*—120° B43 6°) +2, (—20°47 a3 B—7 a B*) 
+a5—7 a* B+13 a? 3° — 33; Soe oer oeseesoosovecevecs (39.) 
which is easily reduced by (34.) to the form 
= v,°(2a'—206+(*)+a,(20°—7 a? B+a 8°) +a5&—3 ap 
+5.a* 8? — 6, eceeeene eervneeeeeos (40.) 
Again, applying the same general formula (37.) to the elimi- 
nation of x, between the equations (34.) and (40.), by making 
now 
p! = —H+4+22 8, g = @?—B, w= —a, J = 1, 
p= a&& —3 aA B45 0° BP —P, g”=2a°—7 a B+ 1,7” >(41.) 
=2a'—2¢° B+/*, 
we find after reductions, 
0 = 25 a8—250 a6 84975 a f° —1850 a'* BF 
41725 a B*—700 uw 654100 a8 BS ... (42) 
that is, 
(jes —— 25 af (a°—2 B)? (a4 —3 a B+ 2°). weevce (43.) 
But this condition cannot be satisfied, consistently with the 
suppositions which we have already made that neither D nor E 
vanishes; because, by expressions similar to (28.), we have 
D = —(a'—3 2° 6+"), E= —aB(#*—2 8). (44) 
We must therefore reject the supposition (30.), and adopt 
the only other alternative, namely, (29.); and hence we have, 
by (9.) 
S(%) = q &yy F(X) = JXoy f (x3) = 7 Xs SF (4) 
= 7X45 (25 — G Use eeeeeaeeseee ( 5.) 
