Similarly, by means of trigonometrical functions, the trough line with central inner conductor can be handled, i.e., it is transformed into a wire of nearly circular cross-section parallel to and in front of a grounded plane, then by means of the bipolar coordinate transformation, the problem of this trough line can be solved by using a finite number of rectangular harnomics.

The purpose of the paper is to introduce a new technique of improving the laser alignment accuracy which is built on the basis that the laser beam drift can he described by trigonometrical functions under the certain conditions.

Based on a comprehensive study on the existing analytical solutions for rectangular plates,a new deflection function is proposed in this paper which consists of trigonometrical functions and polynomial expressions. This function is orthogonal and satisfies all the boundary conditions on four free edges and the conditions at four free corner points.

The distribution functions, like those obtained earlier forn=2, involve only trigonometrical functions of the eigenvalue differences.

The analysis depends on a transcendental equation involving both exponential and trigonometrical functions.

By means of elliptical functions the rectangular line with inner central conductor of circular cylindrical shape is transformed into a coaxial line with circular outer conductor and nearly circular inner conductor, then by employing a finite number of terms of circular cylindrical harnomics we can fit the boundary conditions at the outer conductor and at a finite number of points at the inner conductor, thus the problem of the rectangular line with inner central circular conductor is solved. Similarly, by means...

By means of elliptical functions the rectangular line with inner central conductor of circular cylindrical shape is transformed into a coaxial line with circular outer conductor and nearly circular inner conductor, then by employing a finite number of terms of circular cylindrical harnomics we can fit the boundary conditions at the outer conductor and at a finite number of points at the inner conductor, thus the problem of the rectangular line with inner central circular conductor is solved. Similarly, by means of trigonometrical functions, the trough line with central inner conductor can be handled, i.e., it is transformed into a wire of nearly circular cross-section parallel to and in front of a grounded plane, then by means of the bipolar coordinate transformation, the problem of this trough line can be solved by using a finite number of rectangular harnomics. When the distance between the axis of the inner conductor and the bottom of the trough tends to infinity, the results obtained in this paper go into that of the well-known slab line.

It has been pointed out that the orbital coefficient a of straight chain and monocyclic conjugated molecules is the sine and cosine function of atom number q:a=(2/(n+1))~(1/2) 2 sin ωq a=(2/n)~(1/2) cos ωqwhere the angular frequency ω is related to the energy level E:E=α+2β cos ωThe author in his previous work calls the function a=f(q)the carrier wave of molecular orbit (which is a function where the variable is positive integers). So it is clear that the carrier waves of straight chain and monocyclic molecules...

It has been pointed out that the orbital coefficient a of straight chain and monocyclic conjugated molecules is the sine and cosine function of atom number q:a=(2/(n+1))~(1/2) 2 sin ωq a=(2/n)~(1/2) cos ωqwhere the angular frequency ω is related to the energy level E:E=α+2β cos ωThe author in his previous work calls the function a=f(q)the carrier wave of molecular orbit (which is a function where the variable is positive integers). So it is clear that the carrier waves of straight chain and monocyclic molecules are harmonic waves.The above conclusion is here extended to ordinary conjugated molecules with homogeneous nucleus and identical bonds. The molecular structure is first dissolved into several chains. The atoms joining the chains are joint atoms. It is proved that if the energy level E meets the requirement |(E-α)/2β|<1, then the carrier waves on the individual chains are all harmonic waves with equal frequency, while the relation between the angular frequency ω and the energy level E remains as E=α+2β cos ω. As for the energy levels meeting the requirement |E-α/2β|>1, the carrier waves on the individual chains are not harmonic but catenary or alternate catenary. The latter may be regarded as harmonic waves with imaginary frequencies or those with complex frequencies. This characteristic feature of carrier waves is called harmoniousness.At the critical energy level E=α±2β, the carrier wave loses its harmoneousness, becoming linear wave or alternate linear wave.The orbital coefficients of joint atoms D_1, D_2,…D_n are called joint coefficients. In this paper the equation of carrier waves containing joint coefficient and angular frequencies is derived: α=D_i sin(m+1-q)ω/sin mω+D_j sin(q-1)ω/sin mωThe above equation can be used to calculate the orbital coefficient of the atoms on each individual chain P_iP_j. In order to calculate the joint coefficient and the angular frequencies, the simultaneous equations of joints are also derived.-2 cosωD_i+sum from j=1 to n h_(ij)~(ω) D_j=0 i=1, 2, …n where h_(ii) (ω) and h_(ij)(ω) (i≠j) are called self-related and related function respectively. They are both trigonometrical functions and can easily be expressed according to the geometrical properties of their molecular structures.Let the coefficient determinant of the simultaneous equations of joints be zero, an equation with ω(the angular frequency) as an unknown can be obtained.Solving the frequency equation we obtain the angular frequency, then substitute it into the simultaneous equations of joints, the joint coefficient can be obtained by solving the homogeneous dependent simultaneous equations. Finally, it is substituted into the equation for carrier waves to find the orbital coefficient. If it is substituted into E=α+2β cos ω, then the energy level can be found.If there are only a few joint atoms, then the order in the determinant is low, making it easy to expand, and the simultaneous equations of the joints can be easily solved too. So this method is simple and convenient. Some molecules may have many joints, but they belong to the same category. The difference equation can be used to replace the simultaneous equations of joints. If this method is used together with the difference equation, it would be even simpler.As for heteronuelear conjugated molecules composed of the two different kinds of atoms alternately arranged, the author first establishes the concept of the orbital parameter b. The relation between b and the orbital coefficient a is where K=((E-a_1)/(E-a_2))~(1/2) is called the ratio of magnitude. Here the functional relation b =f(q) between b and q is named the corresponding carrier wave. It is also proved that the corresponding carrier wave and the angular frequency of heteronuclear molecules are respectively the same as those of homonuclear molecules of the same structure (normalization is not considered). The parameter of the former is also equal to that of the latter. So they can be determined through the calculation for homonuclear molecules. This kind of calculation is called the heteronucleus hamonization calculation. As for the calculation of the energy level of the heteronuclear molecules the following equation for hetoronuclear molecules must be used.: E=(a_1+a_2)/2+2βcosω (1+((a_1-a_2)/4βcosω))~(1/2) This equation has been derived by Dong Cong-hao. In this paper the author has developed a now method of proving.

已知直链和单环共轭分子的轨道系数α是原子编号q的正弦函数及余弦函数: α=(2/(n+1))~(1/2)sinωq及α=(2/n)~(1/2)cosωq其中角频率ω和能级E有关: E=α+2βcosω函数α=f(q)为分子轨道的载波(它是整标函数),可见直链和单环的载波是谐波。本文将上述结论推广到一般的纯核等键长共轭分子上。将分子的结构图分解成若干链条,各链条的衔接点处是关节原子,若能级E满足条件|(E-α)/2β|<1,则各个链条上的载波都是同频率的谐波,角频率ω和能级E的关系仍然是E=α+2βcosω。对于满足条件|(E-α)/2β|>1的能级,各链条上的载波不是谐波而是悬链波或交变悬链波。后者可看成虚频谐波或复频谐波。载波的这种特性称为调谐性。在临界能级,E=α±2β,载波失去调谐性,变成线性波或交变线性波。关节原子的轨道系数D_1,D_2…D_n称为关节系数。推导了含有关节系数和角频率的载波公式: α=f(q)=D_i(sin(m+1-q)ω/sin mω)+D_j(sin(q-1)ω/sin mω)利用它可计算出链条P_iP_j上各原子的轨道系数。为了计算关节系数和角频率,推导出关节方程组: -2cos ωD_i+sum from j=1 to n (h_(ij)(ω)D_j=0 i=1,2…n)其中h_(ij)(ω)及h_(ij)(ω)(i≠j)分别称为自相关函数和相关函数。它们都是三角函数,容易根据分子结构图的几何性质写出。命关节方程组的系数行列式等于零,得到以角频率ω为未知数的方程。求得ω后即可逐步求得E。若关节数目少,用此法比较简便。有些分子虽然关节较多但属于同一类型时,可将此法和差分方程法兼用。对于两种原子相间排列组成的匀杂核共轭分子,建立了轨道参数b的概念,b和轨道系数a的关系是其中K=((E-α_1)/(E-α_2))~(1/2)称为辐比。 b和垡的函数关系b=f(q)为相当载波。杂核分子的相当载波及其角频率和同结构图的纯核分子的载波及角频率是相同的(不考虑规格化),两者的关节参数也相等。因此可通过纯核分子的计算求出它们。这种计算法称为杂核纯化计算法。至于杂核分子的能级计算,则需改用杂核分子的公式: E=(α_1+α_2)/2+2βcosω(1+((α_1-α_2)/(4βcosω))~2)~(1/2)去计算。本文给出这一公式的新证法。

The gravitational lens effect is the direct deduction of gravitational deflection of light in general relativity. The first calculation was carried out by Einstein Refsdal has given the detailed calculation about gravitational lens of point source in flat space. His main formulae are given as (1), (2), (3a), (3b) and (4) in the paper, where n - ds/ds - dM; L1, L2 are the luminosity of images and the other quantities are defined in Fig. 1.The gravitational lens effects weren't certainly evidenced, until walsh...

The gravitational lens effect is the direct deduction of gravitational deflection of light in general relativity. The first calculation was carried out by Einstein Refsdal has given the detailed calculation about gravitational lens of point source in flat space. His main formulae are given as (1), (2), (3a), (3b) and (4) in the paper, where n - ds/ds - dM; L1, L2 are the luminosity of images and the other quantities are defined in Fig. 1.The gravitational lens effects weren't certainly evidenced, until walsh et al. discovered the double QSOs 0957 + 561 A, B in 1979. Since the double QSOs have cosmological distance, then we have to consider the effection of the background space time. Formerly, although the Robertson-Walker metric has been taken into consideration in the calculation, it is only used as revising the distance and the whole calculation is carried in flat space time.In this paper, we are only to discuss the formulae of gravitational lens in close space. As for open space, the formulae are similar, if we change the trigonometrical function into hyperbolic function and give sign properly.The standard cosmological model is adopted in the paper. The relationship between the polar angle in Riemann space and red shift Z has been obtained as equation (12) in the paper, Where .R(to) is the cosmological radius of curvature, q0 the deacceleration parameter, Z the red shift.Corresponding to (1) and (2), we have the formulae (16) and (17), its approximation expansion (33) and (34) by means of spherical trigonometrical relationship.As the same, corresponding to (3), we have equation (20) in the paper, where B=4GM/C2r. For the approximation expansion we have equation (35) in the paper, where RM=R/(1 + ZM),Rs = R/(1 + Zs),ZM and Zs are the red shift of lens and image.Corresponding to (14) we have the formula (30) by means of the law of the conservation of energy and the theorem that the plane angles of dihedral angle are equal in spherical geometry.We have the approximation expansion of (30) as (36) in the paper.It is easy to see that when the radius of curvature R-, all of the expansion formulae (33), (34), (35) and (36) evolve to (1) (2) (3) and (34), the ordinary gravitational lens formulae.In general a1+ a2, L1/L2 and xds are known, then we may get the relationship of M and Z from (16), (17), (20) and (30). The relationship is varied with deacceleration parameter q0. If we knew the red shift Z of gravitational lens and its mass M, we should calculate the deacceleration parameter q0. So our work means something of cosmology. It is a pity that the lens of the double QSOs 0957 + 561 A, B is not a point source but a transparent lens. Therefore, we can't calculate directly QSOs 0957 + 561 A, B with these formulae. If we found some lens to be QSOs, it would be easy to use these formulae.