John Van Metre, an Indian trader, penetrated into the northern portion in 1725, and Morgan ap Morgan, a Welshman, built a cabin in the present Berkeley county in 1727.
Rhayader, built close to the Falls of the Wye (whence its name), owes its early importance to the castle erected here by Prince Rhys ap Griffith of South Wales, c. 1178, in order to check the English advance up the Wye Valley.
Seized by the invaders, castle and town were later retaken in 1231 by Prince Llewelyn ap Iorwerth, who burned the fortress and slew its garrison.
Ptolemy Euergetes (247-222 B.C.) rendered the greatest service to geography by the protection and encouragement of Eratosthenes, whose labours gave the first ap proximate knowledge of the true size of the spherical The .
Later writers add nothing to our knowledge, and are chiefly interested in the tarandus, an animal which dwelt in the woods of the Budini and seems to have been the reindeer (Aristotle ap. Aelian, Hist.
': opal e ° °o T A R ple ' ag a ',ap iJ,wl Karkinit A C K r B L Scale, English Miles D S E A 32 Stavropol P O L A PI A N L A s E Derbent ° I?
With very few exceptions only the name AP/AKHI (with various epithets) occurs on the coins of the Parthian kings, and the obverse generally shows the seated figure of the founder of the dynasty, holding in his hand a strung bow.
Hence the sun (according to Mimnermus ap. Athenaeum xi.
The word is Celtic, appearing in Welsh (very frequently) as afon, in Manx as aon, and in Gaelic as abhuinn (pronounced avain), and is radically identical with the Sanskrit ap, water, and the Lat.
Ap. Apical organ.
Dinon ap. Athen.
Ctesias ap. Photium 2-7; many traces of it were afterwards transferred to the story of Ardashir I.
Berossus (ap. Euseb.
Ephorus ap. Strabo, viii.
He has only one symbol (written somewhat like a final sigma) for an unknown quantity, which he calls apc0µ6s (defined as "an undefined number of units"); the symbol may be a contraction of the initial letters ap, as A Y, K Y, D Y O, &c., are for the powers of the unknown (Suvaµcs, square; icu(30s, cube; Svva,uo& va i ccs, fourth power, &c.).
In 1485 Henry, earl of Richmond, disembarked here on his return from France, and was welcomed on landing by Sir Rhys ap Thomas and much of the chivalry of Wales.
1, and Arrian ap. Phot.
Some personal relics of the celebrated Sir Rhys ap Thomas, K.G.
On the execution for high treason of Sir Rhys's grandson, Rhys ap Griffith, but were restored to the family under Queen!Mary.
Julius Hare belonged to what has been called the "Broad Church party," though some of his opinions ap p roach very closely to those of the Evangelical Arminian school, while others again seem vague and undecided.
Other titles of kindred meaning are ap X 77yfLS (" founder") and 7ravaXais, the protectress of the Achaean league.
All symmetric functions are expressible in terms of the quantities ap g in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit.
The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (Xï¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence?
When C vanishes j has the form j = pxg x, and (f,j) 3 = (ap) 2 (aq)ax = o.
Hence, from the identity ax (pq) = px (aq) -qx (ap), we obtain (pet' = (aq) 5px - 5 (ap) (aq) 4 pxg x - (ap) 5 gi, the required canonical form.
Versions ap X ab and aXab), a Hebrew name, mentioned only once in the Old Testament (Gen.
Ap, Anterior lateral process (same as RAP and LAP in fig.
Only the central and horizontal parts of this structure correspond precisely to the entosternite of Limulus: the right and left anterior processes(marked ap in figs.
» III A, Dorsal view; I to VI, HIV the six prosomatic ap -' V pendages.
A well-preserved gateway of red sandstone and portions of two towers of the castle are included in the buildings of the present gaol, and the old parish church of St Peter contains some interesting monuments, amongst them being the altar tomb (of the 6th century) of Sir Rhys ap Thomas, K.G., and his wife, which was removed hither for safety at the Reformation from the desecrated church of the neighbouring Priory of St John.
Thus, while of German periodicals ap p earing in Hungary there were in 1871 only 85, they increased in 1880 to 114, in 1885 to 141; and they were, at the beginning of 1895, still 128, in spite of the constant spread of that process of Magyarization which has, since 1880, considerably changed the linguistic habits of the people of Hungary.
(iii.) (a) If a= then x = aP. (b) If p = log a x, then x = aP. (2) Where the inverse operation is performed with the unknown quantity or number: (i.) If B=A-X, then A=B+X.
It should be observed that, by analogy with the definition of a fraction, a P l q mean (al/q)P, not (aP)llq.
Dafydd ap Ieuan ap Einion held it for the Lancastrians, until famine, rather than Edward IV., made him surrender.
Bouring point P, also self luminous, in the plane of the object, the waves which issue from it will arrive at B with phases no longer absolutely concordant, and the discrepancy of phase will increase as the interval AP 2 " Man kann daraus schliessen, was moglicher Weise durch Mikroskope noch zu sehen ist.
The question is to what amount must the distance AP be increased in order that the difference of situation may make itself felt in the image.
Now since AP is very small, AL' - PL'= AP sin a, where a is the angular semi-aperture L'AB.
According to the standard adopted, the condition of resolution is therefore that AP, or s, should exceed 1X/sin a.
If 2R be the diameter of the objectglass and D the distance of the object, the angle subtended by AP is E/D, and the angular resolving power is given by X/2 D sin a = X/2 R (3) This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.
Then, if Q be any radiant point and Q' its image (primary focus) in the spherical mirror AP, we have 1 1 2cos4) v l + u 'a ' ' where v 1 = AQ', u =AQ, a =OA, =angle of incidence QAO, equal to the angle of reflection Q'AO.
Taking as the standard phase that of the secondary wave from A, we may represent the effect of PQ by cos 27r (_) .ds, where, l = BP - AP is the retardation at B of the wave from P relatively to that from A.
5, speaks of "mare-milkers," and Hesiod, ap. Strabo vii.
The music of the temple attracted the attention of Theophrastus (ap. Porph.
Resolutions, ap. Mansi, op. cit.
2 enclosing a point B, the pressure p at B is the limit of OP/DA; and p =lt(AP/DA) =dP/ dA, (I) in the notation of the differential calculus.
Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.
3 / second, across any curved line AP from A to P, this being the same for all lines in accordance with the continuity.
Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of P to P', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.
(12) Along the stream line xBAPJ, t ' =0, u=ae-" c bl, n; (13) and over the jet surface JPA, where the skin velocity is Q, - q = - Q, u = ae rs Q /m = ae rs lc, (14) ds denoting the arc AP by s, starting at u = a; a ' ch nS2=cos nB= -a' u u - - a b' (15) a l a - b l u - a' a-a' u-b' co > u = ae'" S " c > a, and this gives the intrinsic equation of the jet, and of curvature ds '&1) _ i dw i dw dS2 P= - dO = Q a0 - Q as2 = Q c u-b d (u -a.u -a') _ ?
Rf > a> b> o> a'> -Do; and then so that PT =c/Zir, and the curve AP is the tractrix; and the coefficient of contraction, or breadth of the jet breadth of the orifice - +i' A change of S2 and 0 into nS2 and nO will give the solution for two walls converging symmetrically to the orifice AA 1 at an angle zr/n.