+P=n1, j+k+...+T =n3, It will also be a covariant if the symbolic product be factorizable into portions each of which satisfies these conditions.
Now they could find what really satisfies them and do that.
It has been shown above that a covariant, in general, satisfies four partial differential equations.
Operating with 5l-xidxlwe find S2C 0 =o; that is to say, C ° satisfies one of the two partial differential equations satisfied by an invariant.
This suggests that we should write; la as a l l3; and we find that the use of fractional indices in this way satisfies the laws of integral indices.
(ii.) If x = a satisfies the equation P = o, then poan+plan-1+.
X= a satisfies the equation P = o.
Each symbol a is associated with its supplement a which satisfies the equivalences a+a = i, aa = o, the latter of which means that a and a have no region in common.
To the average man there is a distinction between clothing and ornament, the first being regarded as that covering which satisfies the claims of modesty, the second as those appendages which satisfy the aesthetic sense.
(X.) Hydrostatics Hydrostatics is a science which grew originally out of a number of isolated practical problems; but it satisfies the requirement of perfect accuracy in its application to phenomena, the largest and smallest, of the behaviour of a fluid.
And as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the surface m and satisfies the condition of (6) § 23; so that K is uniform throughout the fluid at any instant, and changes with the time only, and so may be replaced by F(t).
(2) If the actual motion at any instant is supposed to be generated instantaneously from rest by the application of pressure impulse over the surface, or suddenly reduced to rest again, then, since no natural forces can act impulsively throughout the liquid, the pressure impulse W satisfies the equations I do = I d i dos - ax -u, - - y = -v, Pdz = -t, a =p4)-}-a constant, (4) and the constant may be ignored; and Green's transformation of the energy T amounts to the theorem that the work done by an impulse is the product of the impulse and average velocity, or half the velocity from rest.
It is the religious aspect of it which attracts him, the presentation in God of an object which at once satisfies the feelings and the intellect.
Then the position of the loads must be that which satisfies the condition greater than less than an Wi+W2+ ..
Christ suffering on behalf of sinners satisfies the divine righteousness, which was outraged by their sin.'3 His work is an expression of God's love to man; 14 the redeeming power of Christ's death is also explained by his solidarity with humanity as the second Adam," - the redeemed sinner has " died with Christ."
The Bessel function of order m satisfies the differ ential equation dP 2 -}- p - -}- (I _!!) u =o, and may be expressed as the series 2 l dp 2 2 + 2 -?
It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method.
A more subtle danger to which we are especially liable in the case of a dead language is that of our acquiescing in a sense which satisfies us but which would not have satisfied the ancient writer.
According to him, we believe in Nature because it satisfies our material needs, and in God because he satisfies our spiritual needs.
Neither Corinth (as Lightfoot) nor Rome (as Harnack, who assigns it to Bishop Soter, c. 166-174) satisfies all the internal conditions, while the Eastern nature of the external evidence and the homily's quasi-canonical status in the Codex-Alexandrinus strongly favour an Alexandrine origin.
No European language, it would seem, can even boast of a translation which completely satisfies modern requirements.
St Mark's narrative of our Lord's ministry and passion is so simple and straightforward that it satisfies our historical sense.
In this obscure region it is rich in suggestions and rapprochements; but the ingenuity of these speculations attracts curiosity more than it satisfies scientific inquiry.
But it at least satisfies the requirement that the inquiry shall carry the plain man along with it.
Hence the only law of force which satisfies the conditions is that of the inverse square.
Well, say your father has a German valet, and he is a splendid valet and satisfies your father's requirements better than you could, then it's all right to let him serve.
A real number is a class (a, say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of a.
For example, the class of rationals whose squares are less than 2r satisfies the definition of a real number; it is the real number A I 2.
= 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations.
Magnetic induction, like other fluxes such as electrical, thermal or fluid currents, is defined with reference to an area; it satisfies the same conditions of continuity as the electric current does, and in isotropic media it depends on the magnetic force just as the electric current depends on the electromotive force.
Such a view satisfies neither of his interlocutors.
"Philosophy excites and only half satisfies the appetite for truth; I.