The name probably means "very holy" = apt - ayvr,; another (Cretan) form 'Apt67)Xa (_ Oavepa) indicates the return to a "bright" season of nature.
Now D A xA k = (n - k) A k; Aï¿½ A k = k A?1; D ï¿½A A k = (n - k) A k+1;D mï¿½ A k = kA k; (n - k)A ka - w Ak - 1 aA k = O; a _ J (n - k) A k +l A k = O; kA k Ak = wJ; equations which are valid when X 1, X 2, ï¿½ 1, ï¿½2 have arbitrary values, and therefore when the values are such that J =j, A k =akï¿½ Hence °a-do +(n -1)71 (a2aa-+...
The existence of such forms seems to have been brought to Sylvester's notice by observation of the fact that the resultant of of and b must be a factor of the resultant of Xax+ 12 by and X'a +tA2 for a common factor of the first pair must be also a common factor so we obtain P: = of the second pair; so that the condition for the existence of such common factor must be the same in the two cases.
Ukx(n-2) ï¿½ Taking the first polar with regard to y (n - k) (a f) xa x -k-l ay+ k (af) k-l ay -k (ab) (n -1) b12by n kn-2k-1 n-1 k(n-2) =k(n- 2)a u x u5+nax ayux and, writing f 2 and -f l for y1 and 3,21 (n-k)(a f) k+ta i k-1 + k (n - 1)(ab)(a f) k-1 (b f)4 1 k by-2 = (uf)u xn-2k-1?
Then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - ï¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.
(ab), aa, ab, (xa), ax, xx.
X (xa) ki (xb) k2 (xc) k3...axibx2cx3...xx = (AB) hi (AC) h2 (BC) h3...A11 4 13 A1,14131 A B I ?C"' B C "' X (XA) ki (XB) k2 (XC) k3...AXB122cCk...X If this be of order e and appertain to an nie L Eke-/1+2m =e, h i+h2+ï¿½ï¿½ï¿½+221+ji+j2+ï¿½ï¿½ï¿½+kl+li =n, hi+h3+..ï¿½+222+ji+j3+ï¿½ï¿½ï¿½+k2+12 = n, h2+h3+ï¿½ï¿½ï¿½+223+j2+%3+ï¿½.ï¿½+k3+13 =n; viz., the symbols a, b, c,...
For the quadratic aoxi +2a i x i x 2 +a 2 x, we have (i.) ax = 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (ab) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)ax= i'?- (a2 - ao)xix2 - aix2.
4), and issues in a jet between two edges A and A'; the wall xA being bent at a corner B, with the external angle (3= 2Wr/n.
If there are more B corners than one, either on xA or x'A', the expression for i is the product of corresponding factors, such as in (5) Restricting the attention to a single corner B, a = n(cos no +i sin 110) _ (b-a'.0-a) +1!
Generally, by making a' = -oo, the line x'A' may be taken as a straight stream line of infinite length, forming an axis of symmetry; and then by duplica tion the result can be ob A tained, with assigned n, a, and b, of the efflux from a symmetrical converging FIG.
Two corners B 1 and in the wall xA, with a' = -00, and n =I, will give the solution, by duplication, of a jet issuing by a reentrant mouthpiece placed symmetrically in the end wall of the channel; or else of the channel blocked partially by a diaphragm across the middle, with edges turned back symmetrically, problems discussed by J.
The velocity of the ellipsoid defined by X =o is then U= - 2 __ M ((ro b J o (a2 =ab (i -A0), (20) with the notation A or A a a= a (a2bc+ = - 2abc d -- so that in (4) xA x 'UxA x A' 4)' 1 -Ao' (22) in (I) for an ellipsoid.
ZUy2BB0 Bll; reducing, when the liquid extends to infinity and B 3 =0, to = xA o' _ - zUy 2B o so that in the relative motion past the body, as when fixed in the current U parallel to xO, A 4)'=ZUx(I+Bo), 4)'= zUy2(I-B o) (6) Changing the origin from the centre to the focus of a prolate spheroid, then putting b 2 =pa, A = A'a, and proceeding to the limit where a = oo, we find for a paraboloid of revolution P B - p (7) B = 2p +A/' Bo p+A y2 i =p+A'- 2x, (8) p+?
The line XZ consists of a series of lengths, as XA, AB ...
The weight load was pushing downwards, causing pressure on the strut.
67 a) in a fitting position to represent part of the polygon of forces at Xefa; beginning with the upward thrust EX, continuing down XA, and drawing AF parallel to AF in the frame we complete the polygon by drawing EF parallel to EF in the frame.
The various forms in areal co-ordinates may be derived from the above by substituting Xa for 1, µb for m and vc for n, or directly by expressing the condition for tangency of the line x+y+z = o to the conic expressed in areal coordinates.
7) that "he emptied himself and took upon him the form of a servant" (EauTOv µop4 v OovXoD Xa(3c7.v).
Amongst them, actually or potentially, are the grand steward (0yas oircovo,uos), who serves him as deacon in the liturgy and presents candidates for orders; the grand visitor (µryas oaKEAAaptos), who superintends the monasteries; the sacristan (o - KEvocAuAa); the chancellor (X apr041,Xa), who superintends ecclesiastical causes; the deputyvisitor (o rou caKEAAiov), who visits the nunneries; the protonotary (7rpwrovorapcos); the logothete (Aoy06Erns), a most important lay officer, who represents the patriarch at the Porte and elsewhere outside; the censer-bearer, who seems to be also a kind of captain of the guard (Kavarpio-cos or Kavvrp11vQLos); the referendary (pEckpevSapcos); the secretary (i)7rown L uoyp x4wv); the chief syndic (7rpwrEK&Kos), 1 The numbers have varied from time to time.
=x d,b,c d ', c', , d "b" c",, original determinant is = o, and therefore the determinant itself is = o; that is, the linear equations give x'a,b,c - d,b,c =o; a', b', c' d', b', c' a", b', c" d", b", c" which is the result obtained above.