設Γ為一平面曲線而 P 為一定點 , 自P 向Γ所有的切線作對稱點，則所有對稱點所成的圖形Γ1 稱為曲線Γ對定點P 的double pedal curve , Γ1 對定點P 的double pedal curve Γ2 稱為曲線Γ對定點P 的2-th double pedal curve , Γ2 對定點P 的double pedal curve Γ3 稱為曲線 Γ對定點P 的3-th double pedal curve ,…… 。以下是本文主要的結果：結論A：當Γ為一圓形而P 為圓上一點時 , 計算其n−th double pedal curve 的方程式。結論B：當Γ為任意平滑的參數曲線而P 為任意一點時 , Γ的 double pedal curve 的切線性質。結論C：當Γ為任意平滑的參數曲線而P 為(0,0)時, 計算其n−th double pedal curve 的方程式。 Given a plane curve Γand a fixed point P ,the locus of the reflection of P about the tangent to the curveΓis called the double pedal curve of Γwith respect to P.We denote Γ1 as the double pedal curve of Γwith respect to P, Γ2 as the double pedal curve of Γ1 with respect to P , Γ3 as the double pedal curve of Γ2 with respect to P ,and so on , we call Γn the n-th double pedal curve of Γwith respect to P. If Γ is a circle, and P is a point on the circle, we got the parametric equation of the n−th double pedal curve of Γ with respect to P. And, for any parametric plane curve Γ; we got the method to draw the tangent of the double pedal curve of Γ.

We can prove 鬼腳圖 have an one-to-one characteristic; it is mean that you can not design a 鬼腳圖 which will make two starting point to the same end. We also can prove you can design any 鬼腳圖 you want; you can predict a result, and you can design a 鬼腳圖 which suit the result, no matter what the result it is. We can design any 鬼腳圖 we want, but it possibly becomes very big and complicated. We develop a method to make it become briefer. According to the method, we make a function that can design the 鬼腳圖 you want in a very short time. You predict a result in computer, and the\r function will design a 鬼腳圖 which suit the result, and it will be the briefest. 吾人已經可以證明鬼腳圖具備一對一的性質，意思就是：不可能從兩個起點開始畫線，最後到同一個終點上。吾人亦證明：鬼腳圖的結果沒有限定：同一組初始條件可以轉換成任何一組結果。而同一組結果也有許多種不同的畫法，顯示鬼腳圖的畫法不具唯一性。即使如此，畫出來的鬼腳圖可能過於複雜，於是吾人又發展出簡化鬼腳圖的方法，可畫出較簡潔的鬼腳圖。吾人並根據這種化簡方式編出一套程式，只要將欲得的結果輸入，電腦就可以畫出最簡潔的鬼腳圖。

本文發現發放射毛黴（ Actinomucor elegans）為製豆腐乳之最佳菌種。實驗包括形態、生理及抱子萌芽之生理。在靜態合成毛做培養液（ Still synthetic Mucor Solution ）中放射毛黴佳長以第 5 天之乾重量最高，孢子形成與發育在快速生長期之末。最宜生長之生理條件為憑度 30oC 酸鹼度7，相對濕度 92 % ( 30oC 時），振盪培養。更進一步在不同溫度，酸鹼度、濕度、孢子濃度、基本餐養物、氮源、乙醇蒸氣、二氧化碳濃度下加以探討孢子萌芽生理，結果顯示放射毛黴孢子萌芽最宜溫度為 30oC、釀鹼度7，相對濕度 92%，孢子萌芽率因孢子濃度增加而減少，最有效刺激孢子萌芽之基本營養物需要葡萄糖、硝酸鹽和磷酸二氫鹽三種混合物，數種供試的氮源以酵母抽取物刺激孢子萌芽最有效，乙醇蒸氣 1 ％或 l ％以上抱子萌芽即受仰制，大氣中的CO2濃度已足夠孢子萌芽之需，太高或太低濃度反而有害，孢子熱死點測得為55oC。

最古老之動力機械為蒸氣引擎，在早期蒸氣火車發電機皆由蒸氣引擎發展而來，本研究即利用最古老之蒸氣引擎(俗稱飛龍引擎)之製作，以探討內燃機之基本理論以作為模型引擎設計之基礎。在飛龍引擎製作中，歷經引擎無法轉動之痛苦，並極力找尋一百年前發明蒸氣引擎之同樣思考，最後找尋到引擎啟動之合適尺寸，再用其尺寸延伸做實驗，以發展其理論，最後發現引擎設計之重要變數如管長﹑火焰﹑本身重量均為設計中重要因數，最後用正式機械設計方式完成第四代引擎亦正式運轉，可見本蒸氣引擎之設計基礎具有相當實用之基礎。The oldest locomotive engine was the steam engine. All early steam locomotives were developed from the steam engine (Heron engine) and this research project uses the Heron engine to explore the fundamental theories behind the internal combustion engine as a model for basic engine design. During the production stage of the Heron model, there were difficulties with engine rotation, and so efforts were made to find the same thought processes involved in producing the steam engine 100 years ago. The dimension appropriate for moving the engine was first found, and then the idea was extended with experimentation of the size in order to develop the theory of engine design. In the end, the length of the steam hose, the strength of flame, and the actual weight of engine were all found to be important factors of design. Finally, the fourth generation Heron engine was able to rotate properly using the proper engine design method, so it can be seen that the fundamentals of steam engine design has quite a practical engine design basis.

Lantana is a very common plant in our lives. It grows easily and it has a long florescence and various colors. The colors of particular types of lantana alter as the changing florescence. In this experiment, paper chromatography, high-performance liquid chromatography, SDS-gel electrophoresis, the measurement of petal cellular pH values, and the comparative study of forms of trachoma on the epidermal cells of petals are exerted in order to explore factors that change the colors of the lantanaThe findings are as follows:\r (1)Lantana’s colors have inseparable relationships with the compositions of anthocyanins and flavonoids, but not with the pH values of petal cells.(2)The anthocyanins of petal cells are cyanidm, with glycosides as well.(3)Beside the differences in the compositions of pigments, the forms of trachoma on the epidermis of the petal, cone-like or caniniform, can also be used to distinguish different types of lantana, because the trachoma can influence the reflections of light from the epidermis of the petals and also affect colors of the flowers.(4)The result of SDS-gel electrophoresis shows that the biochemical pathways of petal cells in all species of lantana are similar, so we assume that there is mutant in the series of synthesizing enzyme when the anthocyanins of petal cells are formed, and thus, there are no anthocyanins appearing in the yellow and white species of lantanaThe results above are helpful for the understanding and discovering of lantana’s biological mechanisms, and can be used to create new types of lantana and to make further study of the metabolism of lantana’s complete anthocyanin’s biochemical pathway馬纓丹(Lantana ssp.)是常見景觀植物，容易栽種、花期長、花色多，且有些品系花色會隨著花期而變化。本實驗利用濾紙色層分析、高效能液相層析、SDS-gel電泳、細胞pH值測定及花瓣表皮細胞之毛茸(trichoma)型態之比較等方法探討馬纓丹花色之不同及變化的原因。結果顯示: (1)馬纓丹的花色及花色變化與花青素(anthocyanins)和類黃素(flavonoids)之組成有密切關係，而與花瓣細胞內pH值無關。(2)花瓣中所含花青素為矢車菊色素(cyanidm)，並且具有配醣基(glycoside)。(3)花瓣表皮細胞之毛茸型態，如圓錐形或犬牙型，會影響光的反射，進而影響花色，所以毛茸型態可做為區分馬櫻丹品系之特徵。(4)SDS-gel電泳的結果顯示，馬櫻丹各品系的花瓣細胞生合成類似，推測花瓣細胞產生花青素的一系列酵素中，已有突變發生，而造成黃色、白色品系無花青素。以上結果有助於了解馬纓丹花色變化之機制，可將其應用於改良出新的馬櫻丹之品系，或更深入研究馬櫻丹花青素完整生成代謝路徑。

Recently, there have been several researches aimed at the feasibility of electroactive polymers (EAPs) replacing motors as robotic actuators – the driving forces behind mechanical devices. However, current EAP actuators are either hard to control or incapable of discrete accurate movements. The research aims to design a computer interface that makes it possible for the electroactive polymer, VHB 4910, to become an effective substitute for bulky motors in effecting precise and accurate control of a robotic arm.

高二上學期化學第一章，就討論到電子組態及原子光譜，雖然課本上有美麗的光譜插圖，但是觀念還是覺得十分抽象。所以老師為了提高我們學習的興趣，在實驗課中教我們自製簡易分光器，實際去觀賞各種光源及有色溶液的光譜，觀賞過程我們發現下列的問題：（1）為何一般離子是帶狀光譜，而MnO4- 光譜卻像線光譜？（2）MnO4- （深紫）和Mn2+（幾近無色），兩者顏色差異很大。是否與Mn 離子是否單獨存在有關係？（3）而另外CrO42- （黃）、Cr2O72- （橙）和Cr3+（深藍色），三者顏色差異，是否與MnO4- 和Mn2+原因類似？為了尋找這些答案，於是開始了這個題目研究的過程。這期間我們花了很多時間與方法，嘗試將肉眼觀察到的影像，在自設的簡單暗房中，將光柵卡在數位相機的鏡頭前，以腳架或翻拍架拍攝下來。另外我們也應用到高三上學期平衡常數測定實驗中比色法的觀念，以及物理學上單狹縫繞射的觀念，使課本中的理論與實驗研究相互結合！最後我們藉分光光譜儀測定各有色溶液的可見光吸收光譜，再去定量分析這些有色溶液的顏色深淺，並查閱相關的文獻資料。最後發現MnO4-應該是一種電荷傳遞的遷移，所引起特別深顏色的現象。 至於CrO42- 、Cr2O72- 和Cr3+三者顏色比較上是否和MnO4- 和Mn2+一樣的情形？我們推測應該也是如此！初見Cr3+有很深的顏色時，的確嚇了一跳，不過最後我們還是從它們的可見光吸收光譜中發現： CrO42- 、Cr2O72- 在λmax 的吸光度比Cr3+還要大！但是為何肉眼觀察到的Cr3+顏色會比較深？那是因為我們視覺上對藍綠色比較敏感的緣故！請參考表一。 We have discussed some electronic configurations and atomic spectrums in chemistry class. Although there were a few beautiful spectrums in textbook , but it’s difficult to understand. So we made a simple spectroscope by ourselves to observe spectrums of different light source and color solution . After some observation we have found some problems below ： （1）Why the spectrum of color solution is band spectrum , but the spectrum of permanganate ion (MnO4- ) like line spectrum？ （2）MnO4- （deep purple）and Mn2＋（pink）, their color are different very much. Is it because of the lone existence and binding with oxygen atom of manganese ion? （3）Additionally CrO42- （yellow）、Cr2O72- （orange）、Cr3+（deep blue），Are their color’s difference same as MnO4- and Mn2+？ In order to solve it , we started to do the reasrech. We tested different methods much time to record . Finally , We found a good method . That’s placed a grating in front of the digital camera len to take single slit diffraction spectrum pictures in the dark space. Additionally we used spectrophotometer to measure the visible light absorption spectrum. We compared and matched with single slit diffraction spectrums and visible light absorption spectrums. Then we found the intense color of MnO4- due to charge transfer in reference book. How about color’s difference between CrO42- 、Cr2O72- 、Cr3+group and MnO4、 Mn2+ group? We guess they had the same result . We can find the λmax of CrO42- 、Cr2O72- is larger than Cr3+ from the absorption spectrum. But the observation from naked eyes was inverse . This is owing to our vision is more sensitive to blue color. Refer Table 1.

儘管鍺在電子工業上被廣泛運用，但對於暴露在鍺化合物所產生的毒害則尚未被詳細的探討。在探討鍺對細胞所產生的生理影響中，我們使用了二氧化鍺 ( GeO2）和有機鍺（ Ge-132 ）。由實驗結果顯示， GeO2，會造成人類子宮上皮癌細胞（ A 431 ) 及巨噬細胞株（ Raw264.7 ）死亡，而 Ge -132 對細胞生長則不造成任何影響，為了進一步了解鍺引起細胞死亡是否是經過細胞凋亡（apoptosis ) ，我們將鍺處理過的細胞進行染色體 D NA 的分析，結果發現細胞中 DNA 染色體沒有斷裂。由先前 Huang 等人於 1999 年的研究結果顯示，砷對細胞所造成的毒性是經由有絲分裂活化酵素（ MAPK ）傳導路徑，所以為了解鍺誘導細胞死亡的路徑，我們亦分析 MAPK 傳導路徑是否亦參與其中，我們發現 GeO2加入 A431 細胞後，會活化有絲分裂活化酵素中的 ERK ，但對JNK 及 p38 皆無影響，在對蛋白質表現方面，轉錄因子 c-Jun 的蛋白質表現也是隨著GeO2加入的時間增加而上升。 GeO2加入 Raw 264 . 7cell 後，會造成 JNK 、 ERK 的活化，同樣的轉錄因子 c- Jun 也會增加，由此一結果得知鍺對細胞的影響會因細胞的不同而有所差異，為了分析自由基是否參與砷及鍺所造成細胞死亡的過程，我們分析在 A431 細胞中可產生的 NO 的可誘導性 nitric oxide synthase ( iNOS ）的表現，我們發現氧化鍺及砷都會誘導 iNOS 的表現量增加。綜合以上結果，可能顯示氧化錯可能會經由 M A PK 訊息傳遞路徑來促使細胞的死亡，並且 iNOS 亦可能參與此過程。就我們所知，這是第一個提出重金屬所造成的毒害可能會經由 iNOS 來誘導產生的研究。 Despite the extensive use of germanium (Ge) in the electronic industry and optical devices, the potential risks of exposure to germanium compounds have not been evaluated. The effects of germanium on cell physiological functions were studied. We first asked if germanium oxide (GeO2) or carboxyethylgermanium (Ge-l32) could affect cell viability. We found that GeO2, but not Ge-l32, reduced cell viability in a dose-dependent manner in epidermoid carcinoma A43 I and macrophage Raw 264.7 cells. In order to test whether apoptosis contributes to germanium cytotoxicity, DNA fragmentation was evaluated in A43 1 and Raw 264.7 cells treated with GeO2 or Ge-132, respectively. We found that neither GeO2 nor Ge- 132 had effect on chromosomal DNA fragmentation. Previous studies by Huang (1999) et al indicated that sodium arsenite (NaAsO2) cytotoxicity is mediated through mitogen-activated protein kinase (MAPK) pathways. In order to study the mechanism(s) by which GeO2 mediates cell death, we analyzed the signal transduction pathways triggered by GeO2 We found that GeO2 stimulated the extracellular signal-regulated kinase (ERK) activity and transcription factor c-Jun in a time-dependent manner, but not c-Jun amino-terminal kinasc (JNK), or p38 MAPK in A431 cells. Treatment of the Raw 264.7 cells with GeO2, induced activities of ERK, JNK and c-Jun in a time-dependent manner. Collectively, these results suggested that GeO2 effects might be cell type specific. To test whether free radicals were involved in NaAsO2 or GeO2 mediated cell death, the expression of inducible nitric oxide synthase (iNOS), which produced the NO free radical, was determined in A431 cells treated with NaAsO2 or GeO2. We found that expression of iNOS was induced in a time-dependent manner in NaAsO2 or GeO2-treted A431 cells. Taken together, our results indicated that GeO2-induccd cell death may be mediated through MAPK signal pathways and that iNOS may contribute to NaAsO2 or GeO2 mediated cell death. To our knowledge, this is the first report that iNOS may contribute to heavy metal mediated cytotoxicity.

費曼曾說：There is plenty of room at the bottom。喬治亞理工大學的Mostafa El-Sayed 教授發表的癌細胞辨識、與科學月刊報導『台大抗煞一號』引發我們對膠體金粒子的興趣。膠體的性質主要是由界達電位 (zeta potential)決定。參考台科大、成大、中山…等超音波應用研究，提出改良篩選物理法製造之膠體金粒子的儀器設計與製作。經沉降過濾可達平均粒徑 100 nm；而離心式篩選機與超音波管式篩選機可達平均粒徑30 nm。篩選後的膠體粒子以電容原理調控膠體金粒子之界達電位 (zeta potential)，成功地從-30 mV 提升至-59 mV，並發展成電容超音波界達電位控制儀(Capacitor Ultrasonic Zeta Potential Controller)。以膠體金粒子與蛋白質鍵結量來測試調控界達電位的效果，發現蛋白質鍵結量之增加曲線與界達電位的增加曲線的增加趨勢相似；此功能的發現對於生物科技方面的應用應會有很大的幫助。透過界達電位控制系統，本研究達到費曼先生所期望的「在原子或分子的尺度上來加工材料和製造設備」。“There is plenty room at the bottom.” The words of Mr. Feynman are the beginning of nano technology. Mostafa El-Sayed, a professor of Georgia Institute Technology, identified cancer cells through nano gold-antibody complex. So, our study focuses on the zeta potential of colloidal gold particles. At first, the filtering method and equipments were developed. The theories were based on the ultrasonic studies of universities such as National Taiwan University of Science and Technology. Then the colloidal gold’s sizes were filtered to100 nm through settling. At last, by using Continual-Filtering Centrifuge (CoCe.) and Tube Well Mass (TW-MS), the mean particles sizes can be filtered to 30 nm. The most important results are: Zeta potential of the gold colloid was controlled with Capacitor Ultrasonic Zeta Potential Controller. The zeta potential can be raised from -30 mV up to -59 mV, which is -20 mV higher than the conventional pH-changing way. The function of zeta potential to protein binding quantity was tested. The increasing curves of zeta potential and protein binding quantity were similar. This property would be a significance of biotechnology. Thourgh Capacitor Ultrasonic Zeta Potential Control system, the zeta potential’s limitation of gold colliod, which is produced by SANSS (Submerged Arc Nanoparticles Synthesis System), can be controled in a wilder range. The study which is focused on nano-scale, like the wish of Mr. Feynman – “To manufacture material and produce equipment in atom and molecular scale”.

無尾翼飛行器(Tailless Aircraft)在軍事上的價值極大，且對於目前正在起步的微飛行載具(Micro Air Vehicle)而言，亦是值得嘗試與投資的。然而，由於無尾翼飛行器缺乏用以平衡的水平尾翼，造成其靜態的不穩定，即使設法提高靜態穩定特性，但其氣動力阻尼低、穩定性仍舊不佳。操縱上更是困難，在飛行穩定性與控制系統設計上極其挑戰性。本研究目的在探討無尾翼飛行器之穩定性與控制技術，改善其先天之不穩定特性，考慮之項目有縱向靜態穩定性、動態穩定性、控制面與組件配置等因素等進行詳細之探討。首先，找出了適用於無尾翼飛行器之Reflex翼形，接著建立無尾翼飛行器之非線性縱向動態模式，然後針對一翼展8Ocm之小型飛行器進行外型設計，並觀察分析其實際飛行狀態，再以理論與經驗公式估算無尾翼飛行器之氣動力導數，探討其飛行穩定與操控性能。此外，並運用古典控制PID控制法則，設計控制器進行非線性受控系統之動態響應模擬。由模擬結果可看出，經由翼剖面改變與控制系統的輔助下，大幅提高了其性能，使得無尾翼飛行器克服了先天的不穩定特性，更提高了其發展空間 The tailless aircraft has a great value on the military use. Meanwhile, it is worthwhile to try and to invest in it for the investigation of MAV(Micro Air Vehicle), which is being developed now. However, because of lacking horizontal tail which is used for balance, the tailless aircraft is static unstable. Even with the attempt to enhance its characteristics of static stability, the stability of the tailless aircraft is still poor for the sake of it's low damping in aerodynamics. Therefore, it is a challenge to flight stability and control system designing. The purposes of this research are to study the stability and the control technique of the tailless aircraft. To improve its congenital lacking of stability, thought over the longitudinal static stability, dynamic stability and control system. First, find the "Reflex" airfoil is suitable for the tailless aircraft. Second, set up a non-linear and longitudinal dynamic model of the tailless aircraft. Third, design an 80cm span small airplane. Hence, observe and analyze its flying condition. Finally, utilize the theoretical and experiential equations to estimate the aerodynamic derivatives and investigate its stability and controllability. Besides, use the PID controller to proceeded the time-response simulation of the non-linear system. The result of simulation shows that the performance is improved through the change of the airfoil and with the auxiliary of the control system. With this improvement, the tailless aircraft overcome the congenital lacking of stability to broaden its utilization potential.

陸地上的金字塔無時無刻沒有受到氣體的包覆，這個實驗透過風洞模擬金字塔模型，在流體中的壓力變化，並藉由煙線來觀察流體的移動情形，可具體看到當流體經過物體週遭時流場的改變。經由測量壓力，可更深入的探討風速與壓力的關係。在實驗中，可發現當流體迎面而來，在金字塔後方形成的流場變化中，以渦流最易觀察；且透過壓力的測量，發現模型的前、後方，會有明顯的壓差，會對模型造成力矩，可能會使金字塔結構不穩定。越往金字塔上方壓差越小，因此其受阻力所產生的合力矩應較同底面積、同高的長方體小，故金字塔可以長久在沙漠中屹立不搖。此外，流體流過模型兩側會產生分離的情況，分離點的位置會影響渦流區的大小；一般而言，分離點發生在物體的越後方，尾流的尺寸越小，壓差所造成的阻力越低，實驗中發現流速對分離點的位置影響不大，金字塔的擺放形狀卻會對分離點的位置產生明顯的影響，所以越流線型的物體，其受到壓差所產生的阻力越小。希望透過以上的研究，能夠對金字塔周圍複雜的流場有更深入的了解。The pyramid on land is constantly surrounded by gas. Using the wind tunnel, we can investigate the distribution of pressure surrounding the model of pyramid; furthermore, we can discuss about the relationship between pressure and wind velocity. In addition, with the smoke wire, we can visually observe the dynamic flow filed. In our experiment, we discovered that when fluid comes, there will be obvious difference of pressure between the front and the back of the pyramidal model, causing a torque to the model. At higher parts of the model, there is less difference of the pressure, so the resulting torques of a pyramid should be less than that of a cuboid. Therefore, a pyramid can stand in the desert for a long time. Besides, when the fluid flows through the two sides of the model, it will separate, and the position of the separation points will influence the size of the turbulence area. Generally speaking, the farther the separation points are to the back, the smaller the size of the turbulence area, and the less the drag caused by the difference of pressure. With the study above, I expect to have more understanding about the complex flow field of pyramid.

Thermal expansion exists in our daily life. However, thermal expansion is generally too slight to be seen by naked eyes. Therefore, in the present project, a dilatometer was assembled to enhance better sensitivity toward thermal expansion. Hopefully the self-assembled dilatometer could contribute to teaching purpose.The structure of our 4th generation dilatometer is showed below. Using an ‘L’ square to hang up the metal stick and a rolling needle with a mirror to reflect the laser light are the critical parts of this equipment. By using this special reflection mechanism, the slight expansion of a metal stick caused by heat can be enlarged to a large scale. This special mechanism is where our creativity laid. Measuring in millimeter (mm), the measurement precision of the equipment can be extended to 0.0001 decimal. Our dilatometer was used to measure the expansion of various metal sticks caused by the temperature changes. Results were drawn from analysis of the data: 1) The average relative deflection was within 1.0~1.8%; 2) The relative deviation of linear thermal expansion coefficient was within –1.2~-4.4%. 物質熱漲冷縮的特性普遍存在於我們的生活環境中，但因其變化量相對微小，一般並不容易直接觀察，爲了進一步研究這課題，我們組裝偵測熱膨脹的儀器，並希望儀器的靈敏度高，能推廣為教學器材，經過我們不斷努力與改良，終於有了令人愉悅的成果。 自製第四代熱膨脹儀的結構如圖，設計「角尺懸吊金屬棒」與「滾針及鏡面反射」是儀器的重要部份，利用滾針旋轉及鏡面反射雷射光，加乘放大熱膨脹的微量變化，這是我們主要的創意，以公厘（mm）為單位，儀器的精確值到小數第四位。 利用自製的熱膨脹儀，探討金屬熱膨脹的影響因素。分析實驗所得數據，平均相對偏差在1.0~1.8﹪，而線膨脹係數的相對誤差約-1.2~-4.4﹪。