He describes the safe and happy establishment of the people in their own land.
Xl.-lv.) we have no longer a Jewish but a foreign messiah.
+ v ?xl 1 + u l U1+v1V 1 + w1W1) ï¿½ or T x0a,?_ (m-I) J+m(o .
ï¿½; S = r A F(a 0, 711, a2,ï¿½ï¿½ï¿½bo, b l, b 2,ï¿½ï¿½ï¿½9ï¿½ï¿½ï¿½; xl, x2), viz.
For the substitution rr xl =A 11 +1 2 12, 52=A21+ï¿½2E2, of modulus A1 ï¿½i = (Alï¿½.2-A2ï¿½1) = (AM), A 2 ï¿½2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A lï¿½l +ai(A1/.22+A2ï¿½1) +7,2X2/22, A2 = aoï¿½l +2a1ï¿½1/ï¿½2 +a 2ï¿½2 ï¿½ We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a?
=y la xl -i-y2a x2 must also vanish for the root a, and thence ax, and a must also vanish for the same root; which proves that a is a double root of f, and f therefore a perfect square.
A 31 ...n !aï¿½, for a l, a 2, a 3 ...a n respectively; it then becomes a e xi +na l xi -I x 2 +n (n -1)a 9 xl -2 x2 +...
Taking the variables to be x, y and effecting the linear transformation x = X1X+1.11Y, y = X2X+It2Y, X 2 +Y2X Y Xl - X2 y = _ x X I + AI R X 122 so that - ï¿½l b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence.
Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.
In both cases ddl and dal are cogredient with xl and x 2; for, in the case of direct substitution, dxi = cost dX i - sin 00-(2, ad2 =sin B dX i +cos O dX 2, and for skew substitution dai = cos B dX i +sin 0d2, c-&-- 2 n d =sin -coseax2.
In Aristotle the Xl yos of a thing is its definition, including its formal cause, while the ultimate principles of a science are apxal, the "reasons" (in a common modern sense) which explain all its particular facts.'
XL, in which case aA+0B+ ...
Xl., and here Jehovah is six times changed to Elohim, while the opposite change happens but once.
Xl., we have the striking assertion, which surely did not originate in the Temple, that God has no delight in sacrifice and offerings.
It was of the utmost importance to declare that (especially) Isaiah xl.-lxvi.
Was a prophetic work of the highest order; this was reason sufficient (apart from any presumed phraseological affinities in xl.-lxvi.) for ascribing them to the royal prophet Isaiah.
The question of the Isaianic or non-Isaianic origin of the disputed prophecies (especially xl.-lxvi.) must be decided on grounds of exegesis alone.
Xl.-xlviii., and its appendix chaps.
If any other point 0, whose co-ordinates are x, y, z, be chosen in place of 0, as the point to which the forces are transferred, we have to write x1x, yiy, ZiZ for xl, Yi, z1,and so on, in the preceding process.
The relation (44) is equivalent to Xl X~
TABLE XL-Production of Iron Ore (in thousands of long tons).
For the Scottish Excavations see Proceedings of the Society of Antiquaries of Scotland, xx.-xl., and especially J.
Xl.-xlviii., in which Cyrus is the anointed of Yahweh.
[[Table Xl]] - Principal Days of the Mahommedan Calendar.
First comes a general theory of constitutions, right and wrong (Books A,B,F); and this part is afterwards referred to as " the first discourses " Ev Tois rpWTocs Xl yocs).
Count Gunther XL., who died in 1552, was the last common ancestor of both lines.
[Aiy]EL ['1770-ODs 67r[oo idy W(rLV [0, ad E[io ]v i&Beol Kai [b]7rov E[IS] i(rTLV pOvoc, [Xl]yw, iyc. EIpL pET' ailT[OU].
Xl.), Sethites (Cainites) should also here be mentioned, and finally the Carpocratians.