The general type of problem is to find two, three or four numbers such that different expressions involving them in the first and second, and sometimes the third, degree are squares, cubes, partly squares and partly cubes, &c. E.g.
The soldiers lifted the canteen lids to their lips with reverential faces, emptied them, rolling the vodka in their mouths, and walked away from the sergeant major with brightened expressions, licking their lips and wiping them on the sleeves of their greatcoats.
For the most part, the facial expressions of those sitting around the table were sympathetic, but Dulce looked as if she was ready to break into tears.
Language has crystallized them into certain definite notions and expressions, without which we cannot proceed a single step, but which we have accepted without knowing their exact meaning, much less their origin.
Papias actually confounds expressions of Jesus with verses from the Apocalypse of Baruch, referring to the amazing fertility of the days of the Messianic kingdom (Papias in Iren.
He urged on the bill by which Catholics were prohibited from sitting in either House of Parliament, and was bitter in his expressions of disappointment when the Commons passed a proviso excepting James, against whom the bill was especially aimed, from its operation.
When Miss Keller speaks, her face is animated and expresses all the modes of her thought--the expressions that make the features eloquent and give speech half its meaning.
Then he recalled the coarseness and bluntness of her thoughts and the vulgarity of the expressions that were natural to her, though she had been brought up in the most aristocratic circles.
Their expressions were unusually sober.
All wore similar dreamy expressions to Toni's.
On the 25th of November Cromwell charged Manchester with "unwillingness to have the war prosecuted to a full victory"; which Manchester answered by accusing Cromwell of having used expressions against the nobility, the Scots and Presbyterianism; of desiring to fill the army of the Eastern Association with Independents to prevent any accommodation; and of having vowed if he met the king in battle he would as lief fire his pistol at him as at anybody else.
12, 45); though from the lips of slaves and other low persons in the plays we no doubt hear expressions which, while they are quite in keeping with the characters to whom they are allotted, would have shocked the ears of polite society in the 2nd century B.C.
We find in the Meditations no speculations on the absolute nature of the deity, and no clear expressions of opinion as to a future state.
It may be inquired what meaning is to be attached to these expressions, and what are the conditions and the nature of the changes assumed by them.
It consisted essentially in the adoption of Delauny's final numerical expressions for longitude, latitude and parallax, with a symbolic term attached to each number, the value of which was to be determined by substitution in the equations of motion.
The book contains expressions such as daemones, angelica virtus, and purgatoria dementia, which have been thought to be derived from the Christian faith; but they are used in a heathen sense, and are explained sufficiently by the circumstance that Boetius was on intimate terms with Christians.
Ribot, then president of the council, in the Chamber of Deputies, but the expressions he used were so vague that they did not entirely remove the prevailing doubts as to the existence of a formal treaty.
The resistance found from the above expressions includes the components I, 2 and 4 of § 4.
The curves corresponding to the above expressions are plotted in fig.
Literature and affairs, science and statecraft, poetry and medicine, these various expressions of human nature and activity were so harmoniously balanced that they might be found in the possession of one and the same individual.
Its foundation was probably certain expressions lamenting Scottish interference in English affairs.
Before the rise of Neoplaton ism proper we meet with various mystical or semimystical expressions of the same religious craving.
But the exact meaning which he attaches to such expressions is not quite clear; and they occur, moreover, only incidentally and with the air of current phrases mechanically repeated.
It is not surprising that with such maxims as these in his mouth, unguarded in his expressions and careless of his reputation, he should have afforded room for the circulation of many stories to his disadvantage."
AMMONIUS GRAMMATICUS, the supposed author of a treatise entitled IIEpc 6yoLwv Kai &acaopwv (On the Di f ferences of Synonymous Expressions), of whom nothing is known.
With one symbol for an unknown, it will easily be understood what scope there is foradroit assumptions, for the required numbers, of expressions in the one unknown which are at once seen to satisfy some of the conditions, leaving only one or two to be satisfied by the particular value of x to be determined.
Evidence in support of this view is sought for in the accounts in the Anglo-Saxon Chronicle and elsewhere, where the decisions of the witan were received with loud expressions of approval or of disapproval by an assembled crowd, and it is argued that this is a survival from an earlier age, when all the freemen attended the witan.
To treat the actual forms of religion as expressions of our various human needs is a fruitful idea which deserves fuller development than it has yet received; but Feuerbach's treatment of it is fatally vitiated by his subjectivism.
Another singular fact is that they often seemed to be totally unaware of the tendency if not the meaning of some of their own expressions: thus Macleay could write, and doubtless in perfect good faith (Trans.
If we are to follow von Sybel rather than Kugler, this saga of the First Crusade found one of its earliest expressions (c. 1120) in the prose work of Albert of Aix (Historia Hierosolymitana) - genuine saga in its 1 His somewhat legendary treatise, De liberatione civitatum Orientis, was only composed about 1155.
Expressions in Propertius (ii.
It deserves to be recorded as one of the numerous coincidences of discovery that Laplace, on being made acquainted by Lagrange with his new method, produced analogous expressions, to which his independent researches had led him.
Hitherto, in order to prove the equality of two expressions for the same function, it was necessary to transform the one into the other, i.e.
To show that both expressions agreed for every value of the variable; now it is sufficient to prove their agreement to a far less extent" [merely in certain critical points and at certain boundaries].
Two forms of Western Aramaic survive: the Jerusalem form of the dialect, in the Aramaic portions of Daniel and Ezra; and the Galilean, in isolated expressions in the Talmud (3rd century), and in a fragmentary 5th century translation of the Bible.
If a hollow sphere 7 of which the outer radius is R and the inner radius r is placed in a uniform field Ho, the field inside will also be uniform and in the same direction as Ho, and its value will be approximately 3 i - R 3 For a cylinder placed with its axis at right angles to the lines of force, 2 = Ho (41) 2 +4(-2)(i - R2) These expressions show that the thicker the screen and the greater its permeability o, the more effectual will be the shielding action.
We may therefore generally substitute s/2n for tan 0 in the various expressions which have been given for I.
Symbolic Identities.- For the purpose of manipulating symbolic expressions it is necessary to be in possession of certain simple identities which connect certain symbolic products.
When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.
We may say that, in the resulting determinant, the element in the ith row and k th column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the ith row of the second, and has the expression aklb11+ak2b12+ak3b13ï¿½ï¿½ï¿½ +aknbin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows.
Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third.
Respectively: then the result arrived at above from the logarithmic expansion may be written (n)a(n x) = (n)x, exhibiting (n) $ as an invariant of the transformation given by the expressions of X1, X2, X3...