In other insects the imaginal disks are less completely disconnected from the superficies of the larval hypodermis, and may indeed be merely patches thereof.
It is obvious that the area of a group of mountains projected on a horizontal plane, such as is presented by a map, must differ widely from the area of the superficies or physical surface of those mountains exposed to the air.
Given the area A of the plane between the two horizontal contours, the height h of the upper above the lower contour, the length of the upper contour 1, and the area of the face presented by the edge of the upper stratum l.h = A 1, the slope a is found to be tan a = h.l I (A - A i); hence its superficies, A = A2 sec a.
Liouville) (Paris, 1850), which contains also Gauss's Memoir, "Disquisitiones generales circa superficies curvas," and some valuable notes by the editor.
M., thus exceeding the aggregate superficies of Germany, France and the United Kingdom, and containing a population of fifty millions.
We have the memoir Allgemeine AuflÃ¶sung, on the graphical representation of one surface upon another, and the Disquisitiones generales circa superficies curvas.
But these seemed very great difficulties, and I have almost thought them insuperable, when I further considered, that every irregularity in a reflecting superficies makes the rays stray 5 or 6 times more out of their due course, than the like irregularities in a refracting one; so that a much greater curiosity would be here requisite, than in figuring glasses for Refraction.
That I never extended the duplicate proportion lower than to the superficies of the earth, and before a certain demonstration I found the last year, have suspected it did not reach accurately enough down so low; and therefore in the doctrine of projectiles never used it nor considered the motions of the heavens; and consequently Mr Hooke could not from my letters, which were about projectiles and the regions descending hence to the centre, conclude me ignorant of the theory of the heavens.
As regards their superficies, the lava-streams differ greatly.
Between ten and eleven years ago there was an hypothesis of mine registered in your books, wherein I hinted a cause of gravity towards the earth, sun and planets, with the dependence of the celestial motions thereon; in which the proportion of the decrease of gravity from the superficies of the planet (though for brevity's sake not there expressed) can be no other than reciprocally duplicate of the distance from the centre.