The Preparation of Monthly Streamlined Charts Citation Stone, William Welch, Jr. and Weir, Gordon Bruce (1941) The Preparation of Monthly Streamlined Charts. Master's thesis, California Institute of Technology. doi:10.7907/ybrt-x588. https://resolver.caltech.edu/CaltechTHESIS:02192025-225321818 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Meteorology) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Meteorology Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1941 Record Number: CaltechTHESIS:02192025-225321818 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02192025-225321818 DOI: 10.7907/ybrt-x588 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17012 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 21 Feb 2025 23:02 Last Modified: 21 Feb 2025 23:05 Thesis Files PDF - Final Version See Usage Policy. 6MB Repository Staff Only: item control page

THE l?RE:2ARA'l1 ION OF lliO:N .::HLY STrtEAML I HE CHARTS by William v. Stone Jr. and Gordon B. ,veir Submitted in partial fulfillment of the requirements for the degree of Ma ster of Science in Meteorology California Instit ute of Te chnolo gy Pasadena , California 1941 .AC Kl1" OWLEDG EL.IB;NT S The men engaged in the research described in this thesis wish to express their grat itude to the staff of the Meteorology Department of the California Institute of Te chnology of Pa sadena, for their assistance in this problem. Appre ci- ation is also due the Weather Office of March :B1 i eld and the Los Angeles office of the U. S. Weather Bureau for the use of their upper wind charts as sources of data. PRE:?.AR.A TION OJ.P MONTHLY STREAMLINE CH.ARTS Introducti QQ With the establishment of long range forecasting as a definite possibility, there has been created a need for some means of picturing the mean transport of air masses over a longer period of ti me than that re presented by t he daily synoptic chart. In answer to this need, there have been developed mean monthly streamline charts, which show the mean net flow of air at a given level over a period of a month. IG is the purpose of this thesis to describe in detail the construction of these charts,. and to give some indication of the methods employed in their interpretation. A streamline chart is drawn in such a way that at any g iven point on a streamline the resultant tran sport of mass for the month in question is in the direction indicated by the streamline. Through the use of a s treamline analysis as described in this pa per, it is po ss ible to locate zones of deformatj_on, convergence and divergence, and c yclonic and anticyclonic curl. These so-called "centers of action 1' are the elements in which t he long ra nge forec as ter is i ntere s te d , since their p osition at any time in rela- - 1 - tion to their average or normal positions, help explain weather anomalies for the month, and give some indication of the anomalies to be expected in the following month. In order to quickly obtain a rough streamline picture, one may merely plot resultant winds for a given month, and draw the streamline s s o they are tangent at every p oint to the direction of t h e wind at that point. f his method i s not as accurate, however, as that involving an iso gona l analysis, and this paper is concerned only with the drawing of streamline charts by means of an isogonal proce s s. Throughout the discussion in this thesis, it must be rememb e red that the f inal result obtaine d is not an inst antane ous p icture, n or is it an average one. It is a resultant picture obtained by using vectorial resultant values for the winds involved. Choice and .S ource of Data In choo s ing a ba s e level for data in an analys is of this type, it i s desired that a wind level be s elect ed such that the results obtain ed show as much convergence, divergence , curl, etc. as possiule. At the same ti me, however, the level must be high enough t ha t purely local low-level orographic al effects are mj_nimurriized, since each p oint of observati on represents conditions for many square miles on each side - 2 - of it. Before a wind level was selected for use in this thesis, streamline charts were drawn for surface level, and f or a number of' higher elevations. As would be expected, the surfa ce chart showed strong centers of action, but so distorted by surfa ce orographical effects as to give an inaccurate if not erroneous picture. In one instance, surface effects were sufficient to give the wrong sign to the curl of one of the principal centers of action. Ext remely hieh level winds showed so little convergence and divergence, curl, etc. as to be virtually worthless . .b'rom a consideration of the boundaries establi shed by this inve s ti ga tion, a level of approxi ma tely 2000 feet above the s tation was chosen for the taking of wind data. 1hus a t Miami the wind was ta ke n at an eleva t ion of 2000 feet, and at Cheyenne, at 10,000 feet. This level may not be an optimum, but it is such as to eliminate minor orographical effects, while at the same time preserving effects due to large feature s such as the Rocky Mountains, which affect large masses of air. Following is a list of the stations use d as s ources of data, together with the elevation at which the data were taken for each station. - 3 - Seattle, Wash. 2000 1 Al buque ryue 6000 1 :Portland, Ore. 2000' El :Pas o, ir ex. 6000 1 Medford, Ore. 2000' Winnipeg , Can. 4000 1 Redding, Ca lif. 2000 1 Bismarck, No. Dak . 4000 1 San Francisco, Calif. 2000 I North Pla t te, Neb . 4000' Fresno, Calif. 2000 1 Amarillo, 'i1 ex . 6000 1 Burbank, Calif. 2000 1 Del Rio, Tex. 2000 1 San Diego, Calif. 2000' Fargo, No. Dak . 2000 1 Spokane, · ·ash. 4000' Omaha, Neb. 2000 I :Pendleton, Ore. 4000 I Kan. City, Tufo. 2000 I Wirmimucca, Nev. 6000 1 Wichita, Kan. 2000 I Reno, Nev. 6000' Okla. City, Okla. 2000' Las Vegas, Nev. 6000 1 Fort Worth, Tex. 2000' Missoula, Mont. 6000 1 Abilene, Tex. 4000' Boise, Idaho 6000 1 Houston, Tex. 2000 I Modena, Utah 8000 1 Brownsville, Tex. 2000 I Havre, Mont. 6000' St. Paul, Minn. 2000 1 Billings, Mont. 6000' Moline, Ill. 2000 1 Rock Springs, "iy. 10000 1 St. Louis, :Mo. 2000 I Salt Lake, Utah 8000' Memphis, Tenn. 2000 1 Winslow, .Ariz. 8000 1 New Orleans, La. 2000 I Tucson, Ar iz. 4000' Chicago, Ill. 2000' :Phoenix, .A riz. 4000 1 Evansville, Ind. 2000 ' Cheyenne, Wy. 10000' Nashv ille, Tenn. 2000' Denver, Colo. 10000 1 Mont g omery, Ala. 2000 I - 4 - 2000 ' 2000' ~.c. Charleston, s.c. Columbus, Ohio 2000' Jac ksonville, Fla. 2000' Knoxville, Tenn. 2000' Burlington, Ve. 2000' Atlanta, Ga. 4000' Albany, N.Y. 2000' Tampa, Fla. 2000' Boston, i\Iass. 2000' Miami, Fla. 2000' La Guardia Fld., N.Y. 2000' Key West, .l:P la. 2000' Washington, Buffalo, N.Y. 2000' Richmond, Va. 2000' l? ittsburgh, Pa. 2000' Indianapolis, Ind. 2000' Kyle rt own, Pa. 4000' Sault bt. Marie, Mich.2000' Greens b oro, H.C. 2000 1 Caribou, :Maine Pensacola, }fla. 2000' :Detroit, Mich. Spartansburg, ~.c. 2000 I 2000' 2000' It will be note6 that virtually every station making pibal observations was used as a source of data. ihe few stations which were omitted were those which were so close to other stations as to make recording of their data practically a duplication. In addition to the elimination of orographical effects, it is also desired that diurnal effects, such as monsoon winds (in coastal regions) and drainage effects (in mountainous regions ) be eliminated as far as possible. In order to do this, data were taken for both 3:oo .b. M. and 3:00 P. M. PST, times when diurnal e f fects are at a maximum or near-maximmn in both - 5 - positive and negative directions, thus tending to cancel in a monthly picture. Lata were obtained from upper wind charts of the U.S. Army Air Corps and the U.S. Weather Bureau. Army charts were obtained from the Weather Office at NJ.8. rch Field, California, and any missing data such as the first four months of 1941 were supplied through the courtesy of Mr. French, of the Los Angeles Office of the U.S. Weather Bureau. More accurate data may be obtained through the recording of ev.ery pibal ob s ervation directl y from the teletype sequences, since in this wa y errors due to plotting and readin g may be eliminated. However, since time was an important element in this investigation, data were taken from the upper air charts, rather than from the teletype sequences. An important addition to the data in future years would be the inclusion of as many Canadian stations as possible. In this way exa ct loca tion of the Polar front in summer, and an Arctic or secondary Polar front in wint er might be possible. On the charts included in this thesis these fronts may b e f ound only in an approximate manner through an unwarranted extrapolation of ava ilable da ta. - 6 - Recording of Data Recording of the dat a is by far the most tedious po r tion of the entire s treamline analy s is, since da t a must be r e corded f or s ome seventy five stations, twice a day, for a month. As will be seen later, it is neces- sary that t he winds be resolved into north-south and we s t-east comp onents. Just as a vector is resolved into its comp onents, so is t he wind resolved into its components and recorded, all north and west components being recorded a s positive, and all south and east c omponents being recorded as nega ti ve. The table below show s how the dat a were entered for this investigation. Station A Date N w .AM -6 + 4; 1 l?M 1- 9 2 AM PM .• +7 -3 • • +16 -7 -5 • • . • 31 5 • • • -1 AM -t- l?M - 8 +8 Station B Stat ion C w - 7 - 13 N +6 -f 6 - 4 N -0 • ~9 +12 t- 8 • • • • +0 + 6 -1 - 7 - 8 1-- 5 ..J- 7 . - 5 • V{ -6 -1- 0 +7 - 9 . .• • + 5 -+ 10 - 4 No. entries at each station Sum of wind com onents (vector ) Net c omponent in miles per hour At the bot t om of each column is entered the number of entries, the vectorial total of the components, - 7 - and finally the net component, which is the total of each column divided by the number of entries. It is to be noted that when the wind has but one component (i.e. blowing directly f rom the north, south, east, or west), the second component is entered as zero, and is counted as an entry. In this way, for a given station, there should be the s ~me number of entries for a month in both the north-south column and the west-east column. Teams f or the recording of da t a were compo sed of three men: a reader, a computer, and a recorder. The reader reads the st Btion, the .quadrant from which the wind is blowing , the angle the wind direction ma kes with a north-south line (to the nearest 10 de grees ) , and t he velocity in miles per hour. M I JIJ I I s Example B (Abilene) E:rnii,1e A (Seattle) In the examples above, (A) would be read 11 Se attle, north-east, 30-16 11 , and ( B) would be read "Abilene, south-west, 60-18 11 • An accuracy of ten de grees in reading is probably not justified, bec a use of errors in plotting, and bec 2use readers tend to read to angles of 60 degrees ; 30 degrees, and 45 degrees. - 8 - The computer now ta k es this informat ion, and with the aid of Table I, resolves the wind into its components. Table I is merely a sine-cosine table so arranged that for any given angle with the north-south line, and for any velocity up to and including fifty miles per hour, the f irs t half of the column g ives the northsouth component and the s econd half t he we s t-ea s t comp onent. l he computer then read s the comp onents to the 11 r e corder, always g ivi.ng the north-south component first. The recorder enters the numbers given him by the computer in the correct columns on the data sheet, obtaining the correct signs from having listened to the quadrant read by the reader. In t h e examples cited, Seattle would be entered as a positive 14 in the northsouth column, and a negative 8 in the west-east column. Similarly Abilene would be enter ed as a ne gative 9 in the north-south column, and a positi.ve 16 in the westea st column. A table such as Table I is us ed in prefer- ence to a g ra phical method such as Table II, as it pr oved faster and yielded more uniform comp onents f or a g iven wind direction and velocity. To complete the data for one month required ap-prox imatelv" ten hours for a team of three me n, a lthough this time was s omewh at reduced as readers became accus- tomed to the sequence of stBtions, with their various - 9 - elevations, and computers had somewha t memorized Table I. It is possible to operate a two man team if either the reader or t he recorder has memorized f able I completely. By the ti me this investigation was completed, several of the men had memorized the table, thus enabling the completion of data sheets to be accomplished in a much shorter time. Computing o f ~ Teams of two men were required for the computing of the data, one man reading data and recording, and the other man computing on a Marchant calculator. All positive values in a column are added, then from these are subtracted the negative values. '11he resulting total is then divided by the number of entries in the column, and the quotient, which is the mean resultant, is entered in the proper position on the data sheet. It required two men a pp roxima tely four hours to compute the data for one month. Drawing of Isoboreales an~ Isoaustrales When the mean resultants have been computed, they are then plotted on conforma l conic projection maps, the north-south vslues on one map, and t he west-east values on another map. On both maps lines are now drawn through p oints of equal value, much as isobars are drawn, for increments of two miles per hour on both - 10 - positive and ne gative sides of the zero line. These iso-lines on the north-south chart are known as isoboreales and those on the west-east chart as isoaustrales. For ease in reading these lines when the two maps are superimposed (see n rawing of Isogones), the isoboreales are drawn in blue, and the isoaustrales are drawn in red. In the drawing of these iso-lines, it is necessary that they be drawn to con form exac t ly to the data, and t hat no data be disregarded. ~his must be done, since even a minor deviat:i.on which appears on the mean monthly picture may have a considerable significance, and any adjus tment of t he data i s impossible, since one cannot determine in which direction the correction should be made. One mus t pay particular attention to the exact placing of the zero iso-lines, since it is the intersections of the zero isoboreales and zero isoaustrales which determine the position of the centers of action. Becaus e data are available only for the continental United Stat es, the isoboreales and isoaustrales s hould not be extended more than one inch beyond either east or west coastlines, and should never be extrapolated in a direction pa rallel to t he co astline. In Figure I, therefore, the dot ted lines should not be drawn. ~xtra- p olation of this typ e, particula r ly of the zero lines on the west coa st, frequently yields several comp lex - 11 - centers of action which have no r eal existence. In the r eg ion near the coast of the Gulf of ~exico, however, where the p icture is rela t ively a simple one, the isobore ales and isoaustrales may be sa f ely extra polated a cross the Gulf. Dra wing of I sogones .1: he i s oboreale and i s oaustrale maps 8re now 1 stapled t oge t her with a third bl ank map on to p . It is upon thi ~ blank map t hat we draw the isog one s . f he fir s t ste p i s to trace through onto t he blan k map the zero isoborea le s and zero isoaust rales in t he ir app rop riate colors (blue f or isoboreales and r ed for isoaustrales j. An iso g one is by de f ini t ion t h e locus of all conse cut ive po ints whos e a ir transport is in the sa me d irect ion. I sogone s are drawn thr·ough p oints whose north and west compon en t s ha ve ratios of .:t 1 to ± l, ±1 to -:t. 2 , 2nd :!: 2 to :t 1, thus ha ving slopes of' 45, 26½, and 63!- de gr e es 1~es pe c t ively . .l:: 1 or exampl es , see .i!'i gure I I and the mo r e c 0 111pl ica ted ./ i gure VI I. ~Y defin i t ion the zero i s oboreal es and isoa ustra les a re al s o iso gone s. In genera l, there wi ll be t lx e e i sogones to b e d ra wn in eac h quadrant of a cent er of a ction. A ca re ful s tudy of Fi gures I I and VII will g ive t he reader a fa ir concepti on of t he ma ny and - 12 - varied cases which can a.l'lc 6 o occur in the phys ical l')i cture. There are many re f inements which can only be developed by dra wing the cha rts t hemselves , but it i s only by keepin g in m.i.nd the phys ical p icture that confusion can be avoided in the drawing of the iso gones. ~hen one has drawn one or two i s ogonal charts , he i s s ud6en ly confronted with the fact that closed isogone s do exi s t and are very real. They shoul d always be drawn as they are caused by such factors as foehn winds on the lee s ide of mou_rrtain ran ges. I t will be noted later that closed and U- shaped iso gones cause only an i nflection in the streaml ine s ( see F i gure V). In the drawing of isogone s great ca re shoul d be ta ken with the zero i s obore ales and zero isoaustrales , since the curvature of these li nes near p oints where they intersect determines whether or not one will get convergence or dive r gen ce, and of what magnitude. A physically imp ossible p icture may be obt a ined, as in Figure VI, where we have d i vergen ce wit h cyclonic rotation. I n case one obtains this type of picture , the best solution is t o adjust the zerm isoboreales ana zero iso australes so that they do not intersect. Usually the data on the isobore a le and isoaustrale charts will be of such a nature that this adjus tment - 13 - may be made without disregarding any of t he dat a . This di f ficulty will be noted esp ecia lly at t he edge s of the data. .B 're quen tly centers of a ct ion will a ppear along the :Pac i f ic coast, which are not always real, but are due only to the drawing of intersections of the zero isoboreales and isoaustrales, where s uch inter s ec t ions do not actually exist. If one wishes to draw a large s cale and very accurate streamlin·e chart, and the data are good enough, one can dra '-v iso gones through points where the direction of transport has values of .:!: l to .!:"3, .!: 2 to ± 3, i:.3 to ± 2, ±3 t o ..! 1 , e t c . 1: hi s 1 pro c es s woul d n ot b e s imp1 e , a s can be seen by now, and would not be justified unless the da ta were much better than any existing at the present time, but it would undoubtedly g ive compatible result s in any case. :Drawing of .Arrows .QQ Is ogones After the isogones are drawn, small arrows are placed on th em , indicating the wind direction re presented by the isog one. (See Figure VII). It will be noted that the wind direction along a zero isoboreale will be wither due east or west, and along a zero isoaustrale from due north or south. In each case the direction may be readily determined when the isoboreale and isoaustrale charts are superimposed on a light table. - 14 - It will be further noted that the wind d ire ction, that is, the direction of mean net transport, along a zero isoboreale or zero isoaustrale will shift 180 de grees as the zero line passes through a center of action. Drawing of t h e arrows on the isogon es is facilitated i f t he arrow s are dra wn first on t h e ze r o isoboreales and zero i s oaustrales, since the arrow s on isogones between the zero isobore ale and isoaustrale will have orient a tions intermediate between the north-south value on the zero isoaustrale and t h e west-east value of the zero isob ore a le. (See Fi gure VI I). The a r row s are drawn using small triangles with appropriate angular values (i.e. a 45 degree tri angle and a 26½-63½ de gree right triangle). These triangles must oe small, not over 2" on a side, because of the curvature of the coordinate system, and may be easily cut from a sheet of celluloid. Drawing of Streamlines After the arrows are pl a ced on the isog onal chart s, a blank map is stapled on top, and the two charts are placed on a lig ht table. Th e s tre amline s are drawn by simply following the arrows. In other words, wher- ever a streamline crosses an iso gone it must cross in the direc t ion re presented by the arrows on that isog one. In a re gion between two iso g ones the direction of the st r eam- - 15 - line is natu r ally somewhere between the directions of the arrows on the two bounding isog ones. Whenever a stre amline crosses a closed or U-shaped iso gone, there mus t be an inflection in the streamline, as in Figure V. It can be seen that the drawing of the stream- lines is a relatively simple part of t h e process of constructing the final chart, but a good de al of care is necessary in order not to break any of the above rules. This streamline picture is in reality a twodimensional projection of a three-dimensional phenomenon. In a true streamline chart, the gradient across the streamlines is proportional to the velocity along the streamlines. From Figures III, IV, and VI it might be concluded that the velocities at the centers of action would be infinite, but this is not the case, for in order to satisfy the law of conservation, there must be con s iderable vertical motion taking place. For this feason, streamline s need not he carried into the center of 8.ction, but can stop at some distance away from the center. Velocity Profiles The streamline charts by themselves indicate only the direction of the trans port and give no indication whatsoev er of the magni tua e of the transport. - 16 - •J:here- fore the divergence and convergence shown by the chart can only be the conponents normal to the streamlines. It is highly desirable that some way be found to show the other independent comp onent of devergence and convergence--that is, the component along the s treamlines themselves. It is to satisfy this need that velocity profiles are superimposed on the streamline charts. The velocity profiles show one at a glance where divergence and convergence take place along the streamlines, and what the relative magnitudes are. In order to draw these velocity profiles, one must obtain firs,i. the magnitude of the mean resultant wind at each of the stations used as sources of data. This mean resultant wind may be calculated mathematically by squaring the mean resultant north-sou'i.h value at each station, adding to it the square of the mean resultant west-east value, and extracting the square root of the sum. This method is, however, a relatively long and unnecessarily precise method. Sufficiently accurate results may be obtained through the use of a graphical solution as indicated in Table II, in which the abscissa is the absolute magnitude of the west-east mean resultant component, and the ordinate the absolute magnitude of the north-south mean resultant component. - 17 - The magnitude of the mean resultant wind may then be read from the circles of value on the chart. In the plotting of velocity profiles one is i nterested only in the absolute magnitude of the wind, and not in the direction. The values for the mean resultant winds are now plotted on a separate map, and lines of equal velocity are drawn for increments of two miles per hour. The Streamline chart is now superimp osed on the velocity chart, and the iso-velocity lines are traced onto the streamline chart in dotted red lines. It can now be seen that at any point the streamline will give the direction of transport, and the velocity profile will give the magnitude of the transport. In tracing the velocity profiles, it will be noted that inflections of the felocity profiles will occur along lines of convergence and divergence. These in- flections are real and should be made as definite as possible without neglecting any of the data. JPur- thermore, the velocit y profiles should be adjusted so that the zero velocity profile coincides with the various centers of action. As suming continuity of flow and the indestructability of matter , one has in the streamline chart with the velocity pr ofiles superimposed, an excellent representation of a three-di mens iona l phenomenon, that of - 18 - the net monthly trans port of air with its corres pondi nd horizontal and vertical motions. A much more detailed discussion of supplement ary charts, s uch as velocity profile s, vorticity charts, etc. is given i n the pa per by Werenskiold publis h ed in Geofysiske Publikationer Vol. II, No. 9 Av Den Geofysis ke Kommission. Ut,s:;it It is Werenskiold 's method. of drawing streamline charts which was used throughout this entire investigation. Work Ac compli shed During t he third quarter in 1941 at the California Institut e of Technology, the data, s treamline charts, and velocity profiles for the years 1938, 1939, 1 940, and the first four months of 1941 were recorded, computed, and drawn up by a group of seventeen men working for their Master's degree. .rhere was no time avail- 1 able for interp retation of the streamline charts, although this is the most imp ortant part of the analysis as far as long range forecasting is concerned. How- ever, the interpretation will undoubtedly be done in the coming year. The main purpose of this thesis was merely to present a method whic h has been used and which seems to be acceptab le and efficient. - 19 - The research described in this thesis was carried out by the following men: John A. Battle *Sidney c. Bruce *Delmar L. Crowson Darrell Dieter Samuel J. Easley Robert H. Ehrke *Edward A. Hall Daniel S . Hamway Yoshina o 1fakada *Greenup B. Patterson *Isadore I. Porush William H. Rempel Clyde T. Standridge *William w. Stone, Jr. *Bernard M. Tobin *Gilbert R. Van Dyke, Jr. *Gordon B. Weir * Non-flying Cadets in U.S. Army Air Corps - 20 - , ",•10° MPH l 2 3 4 5 I 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 I 49 50 N ., N 1 2 0 0 20° ,I 30° N l I 40° w N 1 0 45 N 1 1 \I \I 50° H· 1 1 2 60° 1i N w 1 1 1 2 2 3 3 1 2 70° i ,t 80° ,N 0- 1 0 2 2 2 2 1 2 0 3 1 3 1 3 2 2 2 2 2 3 1 3 1 4 1 4 1 1 3 3 3 3 3 3 3 3 1 4 1 5 1 5 4 4 3 4 4 4 4 2 5 1 4 6 ~I 5 3 5 4 I 4 4 1 4 5 l· 6 2 6 1 5 7 1 7 2 6 5· 4 4 5 4 5 5 4 6 2 7 1 8 1 7 3 7 4 6 5 6 6 4 6 I 5 7 3 7 1 9 2 8 3 8 5 7 6 6 6 6 7 5 8 3 8 2 10 2 9 3 9 5 8 6 7 7 6 8 5 9 3 9 2 11 2 10 4 10 6 8 7 8 8 7 6 10 4 10 2 12 2 11 4 10 6 1 9 8 9 9 8 9 6 10 4 11 2 2 12 4 11 7 10 8 9 13 9 8 10 7 11 4 12 2 2 13 5 12 rz 11 9 10 10 9 11 7 12 14 5 13 2 15 3 1 4 , 5 13 8 11 10 11 11 10 11 8 13 5 14 3 16 3 15 5 14 8 I 12 10 11 11 10 12 8 14 5 15 3 17 3 16 6 15 9 1 13 11 I 12 12 1113 9 15 6 16 3 18 3 17 6 16 9 14 12 13 13 12 14 9 16 6 17 3 19 3 1 8 6 16 10 15 12 13 13 1 2 15 10 16 6 18 3 20 3 19 7 17 10 15 13 I 14 14 13 15 10 17 7 19 3 21 4 20 7 18 11 16 13 15 15 13 16 11 18 7 20 4 22 4 21 7 19 11 17 14 16 16 14 17 11 19 7 21 , 4 23 4 21 8 20 12 18 15 16 16 15 18 12 20 8 21 4 24 4 22 8 21 12 18 15 17 17 15 18 12 21 I 8 22 4 25 4 23 8 22 13 19 16 1 8 18 16 19 13 22 8 23 4 26 5 24 9 22 13 20 17 18 18 17 20 13 22 9 24 5 27 5 25 9 23 14 21 17 19 19 17 21 14 23 9 25 5 28 5 26 10 24 14 21 18 20 20 18 21 14 24 10 26 1 5 29 5 27 10 25 15 22 19 , 21 21 19 22 15 25 10 27 5 30 5 28 10 26 15 23 19 21 21 19 23 15 26 10 28 1 5 31 5 29 11 27 16 24 20 22 22 20 24 16 27 11 29 5 6 30 11 28 16 25 20 23 23 20 25 16 28 11 30 6 32 33 6 31 11 29 117 26 21 I 23 23 21 26 17 29 11 31 6 34 6 32 12 29 17 27 22 24 24 22 2'7 17 29 112 32 6 35 6 33 12 30 18 27 : 22 25 25 22 27 18 30 112 33 6 36 6 34 12 31 18 28 23 25 25 23 28 18 31 , 12 34 6 37 6 35 13 32 119 28 24 26 26 24 28 19 32 13 35 6 38 7 36 13 33 19 29 24 27 27 24 29 19 33 13 36 1 7 39 ! 7 37 13 34 20 30 25 28 28 25 30 20 34 13 37 7 40 7 37 14 35 20 31 26 28 28 26 31 20 35 14 37 ' 7 41 7 38 14 35 21 31 26 29 29 26 31 21 35 14 38 7 42 7 39 14 36 21 32 27 30 30 27 32 21 36 14 39 7 43 I 7 40 15 37 22 33 28 30 30 28 33 22 37 15 40 7 44 8 41 15 38 22 34 28 31 31 28 34 22 38 15 41 8 45 8 42 15 39 23 34 29 32 32 29 34 23 39 15 42 8 46 8 43 16 40 23 35 29 33 33 29 35 23 40 16 43 8 47 8 44 16 41 24 36 30 33 33 30 36 24 41 16 44 8 48 8 45 16 42 24 37 31 34 34 31 37 24 42 16 4 5 8 49 9 46 17 43 25 35 31 35 35 31 38 25 43 17 46 9 50 9 47 17 4 3 25 38 32 35 35 3 2 38 25 43 17 47 9 l 0 1 l l 2 2 3 1 1 1 1 a I J I I w 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 \\ \ \\ I mph C/;01 t for /fe solv/;1J Jv?!7ds loile f f 0 0- " '' .......... I I -"Do ni,'i e,1,-,(}ow /in .... I cl..""rter).. 0 F;J ][ 0 ~JG YJ1