ABOUT SOME FACTORS DETERMINING THE EFFICIENCY OF FHF WITH INJECTION OF LARGE AMOUNTS OF SAND

Based on the results of laboratory and field tests of formation hydraulic fracturing (FHF) some basic factors are defined which influence upon the effectiveness of FHF technology involved the injection of large quantities of sand into the formation. These factors include a no strictly orthogonal trajectory of FHF fracture; abrasive destruction of initially round perforations; off-center positioning of quashhead shells inside the casing; sphericity and roundness of the proppant (sand) particles, etc.

Transtemporal approach to the removal of a lateral pontine tumor

This video demonstrates a very useful trajectory to a pontine lesion. A 68-year-old man presented with tongue numbness and weakness. The approach used was a transtemporal presigmoid retrolabyrinthine approach to enable an orthogonal trajectory to the lateral pons. Following the transtemporal opening, the root entry zone of the trigeminal nerve and the root exit zone of the facial nerve are identified. The lateral pons is incised to access the tumor, which upon histological analysis was found to be a metastasis. Excellent visualization of the lateral pons is achieved. The opening, relevant anatomy, and closure are illustrated.The video can be found here: http://youtu.be/vS5fCOY6vp8.

III. A supplementary memoir on caustics

It is near the conclusion of my “Memoir on Caustics,” Phil. Trans. vol. cxlvii. (1857), pp. 273, 312, remarked that for the case of parallel rays refracted at a circle, the ordinary construction for the secondary caustic cannot be made use of (the entire curve would in fact pass off to an infinite distance), and that the simplest course is to measure off the distance GQ from a line through the centre of the refracting circle perpendicular to the direction of the incident rays. The particular secondary caustic, or orthogonal trajectory of the refracted rays, obtained on the above supposition was shown to be a curve of the order 8; and it was further shown by consideration of the case (wherein the distance GQ is measured off from an arbitrary line perpendicular to the incident rays), that the general secondary caustic or orthogonal trajectory of the refracted rays was a curve of the same order 8. The last-mentioned curve in the case of reflexion, or for μ = — 1, degenerates into a curve of the order 6; and I propose in the present supplementary memoir to discuss this sextic curve; viz. the sextic curve which is the general secondary caustic or orthogonal trajectory of parallel rays reflected at a circle.

II. A supplementary memoir on caustics

It is near the conclusion of my “Memoir on Caustics,” Philosophical Transactions,, vol. cxlvii. (1857), pp. 273-312, remarked that for the case of parallel rays refracted at a circle, the ordinary construction for the secondary caustic cannot be made use of (the entire curve would in fact pass off to an infinite distance), and that the simplest course is to measure off the distance GQ from a line through the centre of the refracting circle perpendicular to the direction of the incident rays. The particular secondary caustic, or orthogonal trajectory of the refracted rays, obtained on the above supposition was shown to be a curve of the order 8; and it was further shown (by consideration of the case wherein the distance GQ is measured off from an arbitrary line perpendicular to the incident rays) that the general secondary caustic or orthogonal trajectory of the refracted rays was a curve of the same order 8. The last-mentioned curve in the case of reflexion, or for μ = ‒1, degenerates into a curve of the order 6; and I propose in the present supplementary memoir to discuss this sextic curve, viz. the sextic curve which is the general secondary caustic or orthogonal trajectory of parallel rays reflected at a circle.

XIII. A memoir upon caustics

The following memoir contains little or nothing that can be considered new in principle; The object of it is to collect together the principal results relating to caustics in plano , the reflecting or refracting curve being a right line or a circle, and to discuss with more care than appears to have been hitherto bestowed upon the subject, some of the more remarkable cases. The memoir contains in particular researches relating to the caustic by refraction of a circle for parallel rays, the caustic by reflexion of a circle for rays proceeding from a point, and the caustic by refraction of a circle for rays proceeding from a point; the result in the last case is not worked out, but it is shown how the equation in rectangular coordinates is to be obtained by equating to zero the discriminant of a rational and integral function of the sixth degree. The memoir treats also of the secondary caustic or orthogonal trajectory of the reflected or refracted rays in the general case of a reflecting or refracting circle and rays proceeding from a point; the curve in question, or rather a secondary caustic, is, as is well known, the Oval of Descartes or 'Cartesian': the equation is discussed by a method which gives rise to some forms of the curve which appear to have escaped the notice of geometers. By considering the caustic as the evolute of the secondary caustic, it is shown that the caustic, in the general case of a reflecting or refracting circle and rays proceeding from a point, is a curve of the sixth class only. The concluding part of the memoir treats of the curve which, when the incident rays are parallel, must be taken for the secondary caustic in the place of the Cartesian, which, for the particular case in question, passes off to infinity. In the course of the memoir, I reproduce a theorem first given, I believe, by me in the Philosophical Magazine, viz. that there are six different systems of a radiant point and refracting circle which give rise to identically the same caustic. The memoir is divided into sections, each of which is to a considerable extent intelligible by itself, and the subject of each section is for the most part explained by the introductory paragraph or paragraphs. Consider a ray of light reflected or refracted at a curve, and suppose that ξ, η , are the coordinates of a point Q on the incident ray, α , β the coordinates of the point G of incidence upon the reflecting or refracting curve, a , b the coordinates of a point N upon the normal at the point of incidence, x , y the coordinates of a point q on the reflected or refracted ray.

II. A memoir upon caustics

The principal object of this memoir, which contains little or nothing that can be considered new in principle, is to collect together the principal results relating to caustics in plano , the reflecting or refracting curve being a right line or a circle, and to discuss with more care than appears to have been hitherto bestowed upon the subject, some of the more remarkable cases. The memoir contains in particular researches relating to the caustic by refraction of a circle for parallel rays, the caustic by reflexion of a circle for rays proceeding from a point, and the caustic by refraction of a circle for rays proceeding from a point; the result in the last case is not worked out, but it is shown how the equation in rectangular coordinates is to be obtained by equating to zero the discriminant of a rational and integral function of the sixth degree. The memoir treats also of the secondary caustic or orthogonal trajectory of the reflected or refracted rays in the general case of a reflecting or refracting circle and rays proceeding from a point; the curve in question, or rather a secondary caustic, is, as is well known, the Oval of Descartes or ‘Cartesian: the equation is discussed by a method which gives rise to some forms of the curve which appear to have escaped the notice of geometers.