If the coenosteum of Millepora be broken across, each pore-canal (perhaps better termed a polypcanal) is seen to be interrupted by a series of transverse partitions, representing successive periods of growth with separation from the underlying dead portions.
The Daniell type consists of a teak trough divided into five cells by slate partitions coated with marine glue.
Recalling the formulae above which connect s P4 and a m, we see that dP4 and Dp q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10 P 01 4) respectively.
The warehouse was lit, but metal-framed partitions covered in sheeting formed a maze.
Jackson began ripping down the sheets and knocking over the partitions as he yelled for Elisabeth.
This is the typical arrangement, which is exhibited in the majority of the Polychaeta and Oligochaeta; in these the successive chambers of the coelom are separated by the intersegmental septa, sheets of muscle fibres extending from the body wall to the gut and thus forming partitions across the body.
Even so, Prussia was bereft of half of her territories; those west of the river Elbe went to swell the domains of Napoleon's vassals or to form the new kingdom of Westphalia for Jerome Bonaparte; while the spoils which the House of Hohenzollern had won from Poland in the second and third partitions were now to form the duchy of Warsaw, ruled over by Napoleon's ally, the elector (now king) of Saxony.
Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.
Those in the middle are thin, having only the pavement of the cella to support, and are provided with doors and partitions that make a sort of subterranean labyrinth.
The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies.
A partition is separated into separates so as to produce a separation of the partition by writing down a set of partitions, each separate partition in its own brackets, so that when all the parts of these partitions are reassembled in a single bracket the partition which is separated is reproduced.
It is convenient to write the distinct partitions or separates in descending order as regards weight.
Multiplying out the right-hand side and comparing coefficients X1 = (1)x1, X 2 = (2) x2+(12)x1, X3 = (3)x3+(21)x2x1+ (13)x1, X4 = (4) x 4+(31) x 3 x 1+(22) x 2+(212) x2x 1 +(14)x1, Pt P2 P3 P1 P2 P3 Xm=?i(m l m 2 m 3 ...)xmlxm2xm3..., the summation being for all partitions of m.
Be any partitions of X, respectively, the function isexpressible by means of functions symbolized by separation of X1A2X 3.
Certain partitions of those numbers (vide the definition of the specification of a separation).
The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a, z xPy Q in the expansion of the generating function 1 - ax.
The partitions with one bipart correspond to the sums of powers in the single system or unipartite theory; they are readily expressed in terms of the elementary functions.
We can verify the relations s 30 -a310 -3a 20 a 10 + a30, S 21 - 02100 01 -a 2C a 01 -0 11 0 10 021 The formula actually gives the expression of q) by means of separations of (10P01'), which is one of the partitions of (pq).
The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula.
Every symmetric function denoted by partitions, not involving the figure unity (say a non-unitary symmetric function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz.
The general term in a solution involves the product ao°ai 1 a2 2 ...an" wherein Tr =0, Zs7r s =w; the number of such products that may appear depends upon the number of partitions of w into B or fewer parts limited not to exceed n in magnitude.
The extraordinary advantage of the transformation of S2 to association with non-unitary symmetric functions is now apparent; for we may take, as representative forms, the symmetric functions which are symbolically denoted by the partitions referred to.
If we wish merely to enumerate those whose partitions contain the figure 0, and do not therefore contain any power of a as a factor, we have the generator ze 1-z2.1-z3.1-z4....1-z0.
In general the coefficient, of any product A n A m A 7, 3 ..., will have, as coefficient, a seminvariant which, when expressed by partitions, will have as leading partition (preceding in dictionary order all others) the partition (Tr1lr2lr3ï¿½..).
Remark, too, that we are in association with non-unitary symmetric functions of two systems of quantities which will be denoted by partitions in brackets ()a, ()b respectively.
1111 The Number Of Such Terms Is The Number Of Partitions Of W Into 0 0 Parts, The Part Magnitudes, In The Two Portions, Being Limited Not To Exceed P And Q Respectively.
The circulation of air in any given division of the mine is further controlled and its course determined by temporary or permanent partitions, known as brattices, by the erection of stoppings, or by the insertion of doors in the mine passages and by the use of special airways.
Thus the main features of the Japanese dwelling-house were evolved, and little change took place subsequently, except that the brush of the painter was freely used for decorating partitions, and in aristocratic mansions unlimited care was exercised in the choice of rare woods.
Further divisions followed, and the weakness caused by these partitions was accentuated by a rivalry between the two main branches of the family.
In his process a current was passed through a tank divided into two or three cells by porous partitions, hoods and tubes were arranged to carry off chlorine and hydrogen respectively, and the whole was heated to r20° F.
Accordingly the Dual State was involved in a common downfall, and in the three partitions of 1772, 1792 and 1795 to which it was subjected at the hands of Russia, Prussia and Austria, Lithuania fell a prey to Russia and Prussia.
The precipitating boxes, generally made of wood but sometimes of steel, and set on an incline, are divided by partitions into alternately wide and narrow compartments, so that the liquor travels upwards in its passage through the wide divisions and downwards through the narrow divisions.
The ventilation of ends is effected by means of brattices or temporary partitions of thin boards placed midway in the drift, and extending to within a few feet of the face.
The influence of the Swabian branch of the Hohenzollerns was weakened by several partitions of its lands; but early in the 16th century it rose to some eminence through Count Eitel Frederick II.
It not unfrequently happens that some of the larger houses have one or more partitions, but these are only posts of the main building left by the builders to support the roof, for the apartments have usually no communication with each other except by water.
When Louis was dying in 840, he sent the imperial insignia to Lothair, who, disregarding the various partitions, claimed the whole of the Empire.
The three carpels forming the pepo are separated by partitions (cl).
The political history of the parts of Saxony left by the capitulation of Wittenberg to the Ernestine line, which occupy the region now generally styled Thuringia (Thuringen), is mainly a recital of partitions, reunions, redivisions and fresh combinations of territory among the various sons of the successive dukes.
The Poles were, together with the Ruthenians, the youngest Austrian nation; the repeated partitions of Poland since the 18th century brought them unwillingly under Austrian rule.
Lithuania and the Ruthenian Palatinates, the spoil of former partitions, continued to be incorporated with Russia.
The method gives all the partitions of a number, but we may consider different classes of partitions: the partitions into a given number of parts, or into not more than a given number of parts; or the partitions into given parts, either with repetitions or without repetitions, &c. It is possible, for any particular class of partitions, to obtain methods more or less easy for the formation of the partitions either of a given number or of the successive numbers 1, 2, 3, &c. And of course in any case, having obtained the partitions, we can count them and so obtain the number of partitions.
Arbogast's rule of the last and the last but one; in fact, taking the value of a to be unity, and, understanding this letter in each term, the rule gives b; c, b2; d, bc, b; e, bd, c, b c, b, &c., which, if b, c, d, e, &c., denote I, 2, 3, 4, &c., respectively, are the partitions of 1, 2, 3, 4, &c., respectively.
We have for the number of partitions an analytical theory depending on generating functions; thus for the partitions of a number n with the parts I, 2, 3, 4, 5, &c., without repetitions, writing down the product I +x.
Observing that any factor 1/I-x l is=l+x l +x 2l +..., we see that in the term Nx the coefficient is equal to the number of partitions of n, with the parts I, 2,, ..,withh repetitions.
+Nxnzk+.., we see that in the term Nx n z k of the development the coefficient N is equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4,, without repetitions.
The formula shows that except for numbers of the form (3n 2 n) the number of partitions without repetitions into an odd number of parts is equal to the number of partitions without repetitions into an even number of parts, whereas for the excepted numbers these numbers differ by unity.
Thus for the number II, which is not an excepted number, the two sets of partitions are in each set 6.
We see that in the term Nx z of the development the coefficient Nis equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4, ..., with repetitions.
The cathammal areas may remain very small, mere wedge-shaped partitions dividing up the coelenteron into a four-lobed stomach, the lobes of which communicate at the periphery of the body by a spacious ring-canal.
This geographical division was not reproduced by Rome in any administrative partitions of the province.
Standing partitions broke up areas of the expanse.
The keeper opens the two center partitions to examine the brood cells.
The living tissues at the surface are cut off from the underlying dead portions by horizontal partitions termed tabulae, which are formed successively as the coenosteum increases in age and size.