On integration, log(I/Io) = - hA4 x..
The first (a mechanical proof) begins, after some preliminary propositions on the parabola, in Prop. 6, ending with an integration in Prop. 16.
In addition, he wrote a number of scientific memoirs and papers, including two on the integration of partial differential equations (Jour.
Laplace owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until Lagrange showed him the way.
The integration of the scattered tribes of Arabia in the 7th century by the stirring religious propaganda of Mahomet was accompanied by a meteoric rise in the intellectual powers of a hitherto obscure race.
By the principle of superposition the whole effect may be found by integration of the partial effects due to each element of the surface, the other elements remaining at rest.
All that it is necessary to assume is that the effects of the successive zones gradually diminish, whether from the increasing obliquity of the secondary ray or because (on account of the limitation of the region of integration) the zones become at last more and more incomplete.
According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular.
This integration was originally employed by P. Kelland.
Accordingly, the optical distance from AoBo to A is represented by f (A +S/c)ds, the integration being along the original path Ao.
If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§ 3) by merely limiting the integration to the transparent parts of the aperture.
The integration can be effected without much difficulty.
The integration of the several terms may then be effected by the formula e y dy =r(4+2)=(4 - i)(4-2)...
Cauchy, without the use of Gilbert's integrals, by direct integration by parts.
18), the nearest point on the wave-front, is wholly intercepted, and on the left the integration is to be taken from s = CA to s = co.
The occurrence of factors such as sin 4), or 2 (1cos 0), in the expression of the secondary wave has no influence upon the result of the integration, the effects of all the elements for which the factors differ appreciably from unity being destroyed by mutual interference.
The integral equation of continuity (I) may now be written l f fdxdydz+ff (lpu+mpv+npdso, (4) which becomes by Green's transformation (dt +d dz dy dx (p u) + d (p v) + d (p w) l I dxdydz - o, dp leading to the differential equation of continuity when the integration is removed.
In elliptic integrals, the amplitude is the limit of integration when the integral is expressed in the form f 4) 1% I - N 2 sin e 4) d4.
(21) where the integration extends throughout the whole space unoccupied by conductors.
We have by partial integration ff1 fV dd - ' 2 dy JJ dx y JJ y dxd dz = V - d dzdxd dz, and Itwo (similar equations in y and z.
Since the change of energy is independent of the path, the finite change between any two given states may be found by integration along any convenient path.
The direct integration of this equation requires that L and v" - v' should be known as functions of p and 0, and cannot generally be performed.
The volume of the briquette for which u is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases with which we are concerned) be performed in either order.
He published a memoir on the integration of partial differential equations and a few others, which have not been noticed above, but they relate to subjects with which his name is not especially associated.
Universally as an " integration of matter and dissipation of motion," and yet mental, social and moral developments are also called evolution, so that, in accordance with the definition, they are also integrations of matter and dissipations of motion.
But here he becomes hopelessly inconsistent, because he had already said, in defining it, that " evolution is an integration of matter and concomitant dissipation of motion " (First Principles, § 145).