THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 197 
+ 2a] XZ x3 Ly + Uy Vy UE Ls + a} My Wy; +L} Vy Ly Xs) f 
+ Qa ay Ly Ly Ly + 2DL} L313 Xs; 
. DUP = 2d 24 23 22 +1 2d 24 x, 20,73; + 4273 237,52, 
+4303 22 22 9,4 1222? 22 220,23. 
32. By Newton’s theorem, 
aD, +D, 120+ +++ +D 3074+ 30= 
whence, making n=2, 3, 4 and 5 successively, and re- 
ducing by means of (Art. 30) 
D,=0;and,.50=0; 
we — 
=-135@, D,x=-436, Dy=-3(D.374+ 56) 
=} eps >), and 
D,=-43(D,27 + D363 + 36) =D,D, - 136°. 
33. Applying the same theorem ft by Sire (2), Art. 
28, we find 
i S37 =0, 2a°=5-5Q, F2*=0; 37° =-5E, 
F2°=3 -5°Q?,. Ja’=—0, Ja2=—8-5QH, Fa?=3-5°QS 
Sa — = Sat'=—11. Q7H,. Ja? =3-5'Q', 
Jv*®=13-5QH, Fr*=- 14.55, Fr"=3.5°Q'-—5ES, 
S28 =24.5°OE, Ja"=—17-54Q‘H, Saz¥®=38.5°Q8 
—18-5QE’, Sv"=38-5°Q°E*?, and Ja%=—4.59Q°E+5E*, 
34. We know that, when (a) and (8) are unequal, 
Su w= Sar Zax* — Sarre, 
and that 
Sot at = 44 (Sate Say, 
Or, dropping the subject x, and the symbol 3, we may, 
after Hirsch, write these relations thus: 
=[a][8]-[a+A], and (aa)=4§[a]*- [2a]}. 
Extending the method commonly employed to establish 
these relations we find the following : 
(ay) =[a](8) fv) - [a+4) fy] -[a +9] 06] - [a] [8 +71 
+2[a+B8+y]; 
(a88) =} [a] [A}-2 [a+ 8] [8] - [a] [28] +2[a+ 26}}, 
(aaa) = 4} [a]* — 8[2a] [a] +2[8a]}?, 
(ay) = [a] (A) [7] (8) - [a] [8] fy +8] - [a] (6 +9] [8] 
