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?

10The name probably means "very holy" = apt - ayvr,; another (Cretan) form 'Apt67)Xa (_ Oavepa) indicates the return to a "bright" season of nature.

00Now 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-+...

00then 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.

00(ab), aa, ab, (xa), ax, xx.

00X (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,...

004), 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.

00The stream lines xBAJ, xA'J' are given by = 0, m; so that if c denotes the ultimate breadth JJ' of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q, m=Qc, c=m/Q.

00If 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!

00Two 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.

00The 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.

00ZUy2BB0 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+?

00The line XZ consists of a series of lengths, as XA, AB ...

0067 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.

00The 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.

007) that "he emptied himself and took upon him the form of a servant" (EauTOv µop4 v OovXoD Xa(3c7.v).

00Romani imperii (Hanover and Frankfort, 1612-1614, xa.

00Amongst 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.

00Hruschka's extractor, first brought to public xa cto s.X?

00Now D A xA k = (n - k) A k; AÃ¯¿½ A k = k A?1; D Ã¯¿½A A k = (n - k) A k+1;D mÃ¯¿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, Ã¯¿½ 1, Ã¯¿½2 have arbitrary values, and therefore when the values are such that J =j, A k =akÃ¯¿½ Hence Ã‚°a-do +(n -1)71 (a2aa-+...

00ukx(n-2) Ã¯¿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?

00then 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= - Ã¯¿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.

00X (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+Ã¯¿½Ã¯¿½Ã¯¿½+221+ji+j2+Ã¯¿½Ã¯¿½Ã¯¿½+kl+li =n, hi+h3+..Ã¯¿½+222+ji+j3+Ã¯¿½Ã¯¿½Ã¯¿½+k2+12 = n, h2+h3+Ã¯¿½Ã¯¿½Ã¯¿½+223+j2+%3+Ã¯¿½.Ã¯¿½+k3+13 =n; viz., the symbols a, b, c,...

00For 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.

00oi rrpov, star, and Xa(3€Iv, to take), an instrument used not only for stellar, but for solar and lunar altitude-taking.

00To obtain (i) and (ii) together, we show that the volume of a sphere is proportional to the volume of the cube whose edge is the diameter; denoting the constant ratio by aX, the volume of the sphere is Xa 3, and thence, by taking two concentric spheres (cf.

00Passing to the point Xefa we find two known forces, the load XA acting downwards, and a push from the strut XE, which, being in compression, must push at both ends, as indicated by the arrow, fig.

00We then have the polygon of forces Exaf, the reciprocal figure of the lines meeting at that point in the frame, and representing the forces at the point Exaf; the direction of the forces on EH and XA being known determines the direction of the forces due to the elastic reaction of the members AF and EF,, showing AF to push as a strut, while EF is a tie.

00

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