Newton's Easy Quadratures ``Omitted for the Sake of Brevity'

In the 1687 Principia, Newton gave a solution to the direct problem (given the orbit and center of force, find the central force) for a conic-section with a focal center of force (answer: a reciprocal square force) and for a spiral orbit with a polar center of force (answer: a reciprocal cube force). He did not, however, give solutions for the two corresponding inverse problems (given the force and center of force, find the orbit). He gave a cryptic solution to the inverse problem of a reciprocal cube force, but offered no solution for the reciprocal square force. Some take this omission as an indication that Newton could not solve the reciprocal square, for, they ask, why else would he not select this important problem? Others claim that ``it is child's play' for him, as evidenced by his 1671 catalogue of quadratures (tables of integrals). The answer to that question is obscured for all who attempt to work through Newton's published solution of the reciprocal cube force because it is done in the synthetic geometric style of the 1687 Principia rather than in the analytic algebraic style that Newton employed until 1671. In response to a request from David Gregory in 1694, however, Newton produced an analytic version of the body of the proof, but one which still had a geometric conclusion. Newton's charge is to find both ``the orbit' and ``the time in orbit.' In the determination of the dependence of the time on orbital position, t(r), Newton evaluated an integral of the form ∫dx/x <sup>n</sup> to calculate a finite algebraic equation for the area swept out as a function of the radius, but he did not write out the analytic expression for time t = t(r), even though he knew that the time t is proportional to that area. In the determination of the orbit, θ (r), Newton obtained an integral of the form ∫dx/√(1−x<sup>2</sup>) for the area that is proportional to the angle θ, an integral he had shown in his 1669 On Analysis by Infinite Equations to be equal to the arcsin(x). Since the solution must therefore contain a transcendental function, he knew that a finite algebraic solution for θ=θ(r) did not exist for ``the orbit' as it had for ``the time in orbit.' In contrast to these two solutions for the inverse cube force, however, it is not possible in the inverse square solution to generate a finite algebraic expression for either ``the orbit' or ``the time in orbit.' In fact, in Lemma 28, Newton offers a demonstration that the area of an ellipse cannot be given by a finite equation. I claim that the limitation of Lemma 28 forces Newton to reject the inverse square force as an example and to choose instead the reciprocal cube force as his example in Proposition 41. (Received August 14, 2002) Published online March 26, 2003 Communicated by G. Smith     

论虚概念

把虚概念理解为“所指称的对象不存在于客观世界的概念”，是不对的，将导致把“理想模型”等本该是实概念的概念纳入其中，也把本该是虚概念的如“地心说”中的概念却排除在外。实际上，对象有虚实，属性也有虚实。实对象、实属性，属于实概念；虚对象、实属性，也属于实概念；实对象、虚属性，属于虚概念；虚对象、虚属性，是虚概念。可见，无论对象虚实，只要所指称的对象的主要特征所反映的属性不是客观的概念就是虚概念。     

四硫代钼酸铵与四硫代钨酸铵合成方法改进的研究

对钼酸铵与硫化氢反应合成四硫代钼酸铵、钨酸与硫化氢反应合成四硫代钨酸铵的方法进行了改进。结果表明，采用改进后的方法，反应时间分别为７小时和１１小时。比文献值￣［１１］缩短了许多。产率分别为９１．１％和５２．１％，均高于文献值￣［１］。     

关于高维系统中Poincαre—Bendixson定理的反例

本文给出了在n(n≥4)维系统中,Poincare-Bendixson定理不成立的多项式系统的反例。     

植物药超氧物歧化酶活性与某些性质的研究

应用NBT光还原法对23种植物药材SOD粗提液活性研究表明。供试材料全部达到200units·g~(-1)F·W的水平,13种超过300units·g~(-1)F·W,野山楂高达812.5units。酶的比活,玄参、何首乌位列前茅,超过300units·mg~(-1)protein~-,野山楂、枸杞等都在150units·mg~(-1)protein~(-1)以上。综合分析,何首乌、野山楂、玄参与枸杞等酶活性与比活都高,有开发利用的价值。植物药普遍具有Mn-SOD与Cu·Zn-SOD,本研究首先发现枸杞、宁夏枸杞、沙参和合子草具有Fe-SOD。酶粗提液的热稳定性明显优于纯化酶。     

在油酸钠存在下醋酸乙烯酯──甲基丙烯酸甲酯的无乳化剂乳液共聚

本文研究了在油酸钠存在下醋酸乙烯酯──甲基丙烯酸甲酯的无乳化剂乳液共聚。发现油酸钠加速聚合反应，并测定了胶乳粒子。     

MeV Pb离子在非晶Si中的射程离散──离子束混合和增强扩散的影响

１．０ＭｅＶ２０８Ｐｂ离子在非晶Ｓｉ中的投影射程ＲＰ和射程偏差ΔＲＰ作为注量和温度二者的函数用背散射法进行测定．注量的变化范围为５×１０１３～７×１０１４ｃｍ－２．注入是在室温和ｔ＝－１２０℃下完成的．由由实验所确定的投影射程，射程偏差与注量或温度无关，并且分别等于２９５和７２．２ｎｍ．与ＴＲＩＭ８６的计算结果相比较，发现ＲＰ的偏离为１８％，而ΔＲＰ的偏离为３６％．ＲＰ和ΔＲＰ二者与注量及温度的无关性，排除了所观察到的与ＴＲＩＭ的矛盾是由于注入期间辐射增强扩散或离子束混合效应而引起的解释。     

具有部分自由液面水平管流的理论计算方法

本文对未充满液体具有部分自由液面的圆管内的流动提出一种理论计算方法.理论可以正确预报"脱体点"的位置、流量、自由液面曲线形状等重要物理量.文中还讨论了管直径D、长度L、流体密度ρ、粘度μ及液位高度等参数对流动的影响.文中以光学玻璃连续熔炼系统中起重要作用的联接管内的流动为例作了具体计算,并与实验结果作了比较,理论与实验结果吻合良好.本文给出的理论方法可推广于一般截面形状的具有部份自由液面的管流计算.