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     

两段绞接式联合收割机输送槽的设计研究

本文介绍一种适宜配置在悬挂式联合收割机上的两段铰接式输送槽设计,这种输送槽后段固定,前段与割台连接并与后输送槽铰接,该铰接点为前悬挂割台的支点,借助液压油缸升降,文中对该铰接点位置的确定,以及两段输送槽铰接处的圆弧过渡板设计进行了分析研究,提供了设计方法.     

灰色系统建模的变换方法

本文通过定理的充分必要性的证明给出了寻找提高原始数据序列光滑度的变换的途径, 对获得新的变换具有指导意义, 实例分析和误差讨论充分展示了这一寻找方法非常有效。     

LOGISTICS SCHEDULING： ANALYSIS OFTWO-STAGE PROBLEMS

This paper studies the coordination effects between stages for scheduling problems where decision-making is a two-stage process. Two stages are considered as one system. The system can be a supply chain that links two stages, one stage representing a manufacturer; and the other, a distributor It also can represent a single manufacturer, while each stage represents a different department responsible for a part of operations. A problem that jointly considers both stages in order to achieve ideal overall system performance is defined as a system problem. In practice, at times, it might not be feasible for the two stages to make coordinated decisions due to (i) the lack of channels that allow decision makers at the two stages to cooperate, and/or (ii) the optimal solution to the system problem is too difficult (or costly) to achieve.Two practical approaches are applied to solve a variant of two-stage logistic scheduling problems. The Forward Approach is defined as a solution procedure by which the first sta     

STABILITY CRITERIA FOR A CLASS OF UNCERTAINSYSTEMS WITH TIME—DELAY

Some stability criteria are obtained for a class of uncertain systems with time-delay usingLyapunov functional and analytic techniques. It is easy to check the criteria by making use of theboundedness of the uncertainties.     

煮糖汁汽加热器故障分析及防止措施

本文对糖厂用煮糖汁汽加热器的不锈钢换热管与管板连接焊缝产生的裂纹和换热管横向断裂问题,进行全面的理论分析,並结合具体情况,提出一些可行的防止焊缝裂纹和管子横向断裂的主要措施。     

三价钕与绕丹宁（RACP）及其衍生物二元体系配合物的电位滴定法研究

本文运用电位滴定法与ＴＩＴＦＩＴ程序研究了２９８Ｋ下，三价钕与新型荧光试剂配合物的稳定性，并得出其相应的络合常数。     

Technical Notes ：STUDIES ON THE VALUE OF SECONDARY MARKET TO THE SUPPLY CHAIN

This paper investigates the impact of a secondary market, where retailers can buy and sell excessinventories, on the supply chain. We develop a two-period model with a single manufacturer and tworetailers. At the beginning of the first period the retailers order and receive products from themanufacturer, but at the beginning of the second period, they can trade surplus products betweenthemselves in the secondary market. We investigate the impact of the correlated dependence ofretailers' demand on both the quantity effect and the allocation effect under the secondary market.Lastly,we study potential strategies for the manufacturer to increase sales with the existence of thesecondary market.