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UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 REPORT NO. 68 . 
 
 A SUMMARY OF HIGH SPEED VACUUM TUBE CIRCUIT WORK 
 FOR OCTOBER AND NOVEMBER, 1955 
 
 by Gene H. Leichner 
 
 December 12, 1955 
 
 Technical Report to the Office of Naval Research 
 
 Contract N6ori-71 
 
 Task Order XXIV 
 
 Project ONR NR 0U8 09^ 
 
PREFACE 
 
 The Digital Computer Laboratory of the University of Illinois 
 has recently been examining direct- coupled asynchronous computer circuits 
 of the highest possible speeds. Switching elements consisting of both 
 vacuum tube and transistor types have been considered. This report 
 presents a comparison of calculated circuit operating times with observed 
 times for some representative circuits in the direct-coupled asynchronous 
 class. The study includes the electron multiplier type of vacuum tube. 
 
 Mr. G. H. Leichner is a research assistant working on a three- 
 quarter time basis under the University's contract with the Office of 
 Naval Research on electronic digital computer techniques. 
 
 Work on direct-coupled asynchronous circuits using transistors 
 is reported in other reports. Monthly progress reports issued by the 
 Digital Computer Laboratory contain additional information about these 
 circuits. Copies of these reports are available from the Digital Computer 
 Laboratory. 
 
 R. E. Meaeher 
 
A SUMMARY OF HIGH SPEED VACUUM TUBE CIRCUIT WORK 
 FOR OCTOBER AND NOVEMBER, 1955 
 
 1. Introduction 
 
 In an attempt to make digital computer circuits with operation times 
 in the range of a few tens of millimicroseconds several ideas on the basic 
 
 design philosophy have been developed. The figure which determines the ultimate 
 
 CV 
 circuit speed under on-off conditions is T = — where C is the total shunt 
 
 capacity from a moving point to all non-moving points, (ground primarily) V is 
 
 the voltage swing required, and I is the amount of current available to move 
 
 the point in question. The above formula can be applied to tubes as a figure 
 
 of merit (a small figure is desired) in the following way: choose a plate 
 
 voltage, take I to be the plate current at some chosen grid voltage such that 
 
 rated plate dissipation is not exceeded (zero or some positive bias), take V 
 
 to be the chosen grid voltage plus the cut-off bias voltage, take C to be the 
 
 total input and output capacity of a grounded cathode stage. This figure of 
 
 merit is a time and is nearly independent of the plate voltage chosen. Typical 
 
 values for a few tubes are given in Table I. 
 
 1 
 
 Tube 
 
 T (mu-sec) 
 
 6AH6 
 
 1.8 
 
 E _ = 100 
 g2 
 
 
 6AK5 
 
 2.3 
 
 E _ = 100 
 g2 
 
 
 6AU6 
 
 k.2 
 
 E . = 100 
 g2 
 
 
 6BQ7A 
 
 0.93 
 
 6j6 
 
 1-5 
 
 5687 
 
 1.1 
 
 EFP60 
 
 
 Plate 
 
 0.78 
 
 Z319 
 
 
 Dynode 
 
 1.7 
 
 Plate 
 
 1.2 
 
 Table I 
 
 2 . Speed Cons iderations 
 
 From the figures of merit it would 
 appear that circuits using the 6j6 (T = 1.5) 
 or the 6BQ7A (T = O.93) might be built in 
 the 10 mu.s range. However some other con- 
 siderations must be taken into account. The 
 6j6 will be used as an example. 
 
 One of the basic problems involved 
 in direct coupled vacuum tube circuits is 
 the difference in potential between the grid 
 and plate circuits. Whenever such circuits 
 are cascaded it is necessary either to come 
 back down from the potential of the plate 
 circuit to that of the grid circuit for the 
 next stage or to progress higher and higher 
 through the circuit. The latter choice is 
 
 1. See Section 7> P» 3- 
 
 -1- 
 
usually not feasible in a computer so the first method will be used in what 
 follows. An effective operation time, t , will be computed for a 6j6 
 (T = 1.5) 
 
 1) To insure cut-off in all new tubes the bias in the circuit is usually made 
 two times the bias read from the tube manual. This makes t = 3*0 mu-s . 
 
 2) The total capacity in the circuit is several times greater than that of the 
 output capacity of one tube plus the input capacity of another, as was assumed 
 in computing T for the tubes. A typical total value is 16 \1\1f where the tube 
 only accounts for k unf. This makes t = 12. The capacity is made up as 
 follows: each pin of a tube socket to all others has 1.5 H-u-f, a 2W resistor 
 has 1 nnf shunt capacity, 1W and l/2W resistors have 0.5 u-u-f, a two-inch piece 
 of wire l/U inch from a flat plate has 1 |i|if, etc. 
 
 When tubes are used in a grounded grid manner, that is using the 
 cathode as the input electrode, the driving source must charge the heater- 
 cathode capacity which is usually about 5 M-M-f or more. The GWih is exceptional 
 with 0.8 u-uf making it a possible tube for a cathode driven gate or for a 
 cathode follower feeding only one point. Unfortunately its maximum cathode 
 current rating of 5° 5 nia prevents its use for high speed driving of capacitive 
 loads . 
 
 3) A single circuit usually has more than one moving point, so each point 
 contributes some delay. While the times are not strictly additive, for two 
 identical circuits the time is approximately twice that of one circuit. For 
 such a circuit t _ becomes 2h mu.s . 
 
 h) When tolerances are considered in the usual flipflop circuit the nominal 
 voltage swing at the plate is larger than minimum allowable swing at the grid. 
 This is partly due to the attenuation of the network. To reduce the attenuation 
 a fairly large negative supply is used. The plate voltage of the tube is made 
 as low as possible to reduce the amount of 'bleed down' required. The lowest 
 plate voltage possible is determined by the plate current required for the 
 
 2. See circuit on page h for example. 
 
 -2- 
 
desired speed and the amount of grid current that must be supplied to get it. 
 Usually a cathode follower is required inside the flipflop if more than 1 ma 
 of grid current is required for a plate current of the order of 5 to 10 ma. 
 While it appears that the 'overswing' would not slow the circuit since only 
 a fixed swing is required for operation, the effect still occurs. If the 
 circuit has reached equilibrium in one of its states, then a certain voltage 
 change is required before the moving point enters the operating range for the 
 worst case. The additional time for this change can easily be equal to the 
 ordinary operation time, depending on the amount of overswing. In some circuits 
 the overswing can be limited by diodes or by some other means. However the 
 addition of a diode introduces extra capacity, particularly when a cathode is 
 attached to the moving point. For an overswing equal to one-half the normal 
 swing t becomes 36 mu-s. 
 
 5) In many circuits the current used to charge the shunt capacity is not 
 constant as is assumed in the figure of merit. Usually however the final 
 (smaller) value is used in calculations so that times calculated are on the 
 slow side. In some cases it is possible to use an inductance in series with 
 the load to reduce the current through the load during the charging of the 
 shunt capacity. This is beneficial only in circuits where the moving points 
 are 'driven 1 in both directions since the resonant speed of the inductance 
 with the shunt capacity is usually slower than the desired operating speed. 
 
 6) The use of ordinary series peaking schemes is sometimes considered- These 
 schemes are not of interest in computer circuits since they only serve to 
 produce fast 'edge' speeds with total delays which are equal to or greater 
 than those which would occur without the peaking. 
 
 7) None of the circuits tried so far has had a speed slower than that which 
 is determined by the external circuits including tube shunt capacities. That 
 is, the transit time of the tube has not yet been approached even with plate 
 voltages as low as 25 v. The transit time for a tube with E = 25 V and with 
 cathode-plate spacing of 0.1 cm (this is approximately the spacing for a 
 6j6) will be computed. 
 
 •3- 
 
E = plate- cathode voltage, statvolts 
 
 t = 
 
 -.2 e = electron charge, statcoulombs 
 
 2d m ' 
 
 eE d = plate- cathode spacing, cm 
 
 m = electron mass, grams 
 
 t = 
 
 (2)(1Q- 2 )(9.0^)(10" 28 ) _ J (9.0^)(10" 3 °) 
 (1 + .7T)(10" 1 °)(3.3)(10' 3 )(25) V (0.197)(10 -10 ) 
 
 t = 6o77 x 10" 10 = O.677 mi-LS 
 
 8) The fastest circuits tried so far have been of the type which provides a 
 driver for the moving point in both directions. Circuits which have points 
 that drift toward some supply voltage, such as the plate circuit of an ordinary 
 amplifier when suddenly cut off, are usually not in the 30 mus region. 
 
 3. Calculations for Some Typical Circuits 
 
 In the calculations which follow, certain simplifying assumptions are 
 made. In places where the point of the circuit in question drifts toward some 
 supply voltage an exponential representation for the voltage as a function of 
 time is used. When the supply voltage is large compared to the change involved, 
 a linear, constant current, representation for the voltage is used. The current 
 in tubes is assumed constant, even for triodes. The calculated times are based 
 on square input signals and consist of the sum of shorter component times which 
 theoretically should not be simply added. However for times of the same order 
 of magnitude the error involved for such an addition is not serious. 
 
 The measurements are usually made in racing registers in which the 
 driving signal for one stage is the output signal from a previous stage so that 
 there are no square signals present anywhere in the circuit. Therefore the times 
 from such measurements should be larger than those calculated on a square wave 
 basis. It is felt that the additional effort which would be required to make 
 calculations based on actual conditions in a racing register is not warranted 
 in this investigation. 
 
 -h- 
 
Ordinarily the following symbols will be used in the examples which 
 appear below: 
 
 c - shunt capacity at a given point 
 
 E - voltage at the plate of a tube with respect to cathode 
 
 E - output voltage of a circuit 
 
 Rt - equivalent plate circuit load resistor ) Thevenin 
 
 ) Equivalent 
 E - equivalent plate supply voltage ) Circuit 
 
 E _ - voltage from control grid to cathode 
 gl 
 
 E - voltage from screen grid to cathode 
 
 Example I . 
 
 Cathode Follower Feeding Two Grids 
 
 IN o 
 
 0, 4/^MsT 
 
 5/j/^t ea. 
 
 A.lju.^ 
 
 OUT 
 
 for. ^/\ch 
 
 HALF OF 6J& 
 
 W f = 
 
 L5"w, 
 
 E P - 
 
 IOO v. 
 
 I f - 
 
 15" ma,, 
 
 1 ., 
 
 % 4(£)-2w. 
 
 R.GSISTOR5. 
 
 Itf =-2.? ma, go,. @+-2v. 
 
 4.0/^f 
 
 £.5 W\C^ 
 
 -|OOv, 
 
 RESISTORS 2^u^f> 
 
 wine \yt/"* J 
 ^ total. ~ ' 5* ^^y^r 
 
 Calculations : 
 
 Assume a 10 v downward swing is required from an upstate voltage 
 of +2 v. 
 
 T - *1 + *2 
 
 -5- 
 
t = S x 2 = (13.5)(2) m lal 
 
 av 
 
 27-5 
 
 mu.s 
 
 = C^c_8 = (15.5)(8) _ 
 t 2 25 25 - >.U mus 
 
 T = 6.1 mu-s 
 
 Measurements : 
 
 This circuit is sufficiently fast that a pulse could not easily he 
 made with much faster edge speeds to test it. 
 
 Example II. 
 
 I'f&f^F 
 
 IN O- 
 
 t^lOy 
 
 + \@, 
 
 ~ro cur ope 
 
 
 Half of a Symmetrical Flipflop 
 Feeding a Cathode Follower 
 
 1 Mi. t z'A 
 
 lw.- 
 
 * % Lfe4= 
 
 2 'A 
 
 
 Calculations : 
 
 Worst case output conditions +1 v at 1 ma, -10 v at ma. 
 Slowest direction of movement is up. 
 
 -6- 
 
E,_ = +89 v 
 beq 
 
 Nominal down voltages E = +25 V, E ■ -17 v, R L = 13- 7K, 
 
 Note that a 7 v overswing occurs, over one-half the 11 v worst case 
 
 swing. 
 
 Equivalent circuit during driftup: 
 fe*CF7.2)||s-M«l£.9* 
 
 E-+3Zv 
 
 -o«£-o ^\A- 
 
 — l~\7« 
 
 T 
 
 c 
 
 T<»TM_- W.^jyuJr 
 
 5\ .* t 
 
 t = (.46)(11.9)(12.9) 
 
 t = 70. 5 mu.s. 
 
 Measurements: 
 
 This circuit was not operated as shown. See Example III. 
 
 Example III. 
 +90 
 
 /I 
 
 5siS~± 
 
 N O i 
 
 \.Sjy* 
 
 Symmetrical Flipflop 
 + sS ° with "Grid" Gate Feeding Cathode Follower 
 
 * -MS"C> 
 
 + 9o 
 
 0V TO 
 CUT OFF 
 
 '< OCKET5 3/-/«.-f } 
 
 w 1 R^ L/y^" J 
 
 lie 
 
 -0 in 
 
 
 -f- 
 
 -7- 
 
 
 1/y^ 
 
 o£, 
 
 7^-r 
 
Calculations : 
 
 Worst case output conditions +1 v at 1 ma, -10 v at ma. 
 
 Slowest direction of movement of E is down. 
 
 o 
 
 Nominal down voltages E = +25 v, E = -17 v, R L = 13.7 K, 
 
 E v = +89 v. 
 beq 
 
 The calculations will be made as follows : 
 
 Assume E is up (+1 v) and the left-hand grid is down. The time re- 
 quired for the flipflop to hold is the time, t, , for the left-hand flipflop 
 grid to reach +1 v plus the time, t_, for the left-hand triode to pull E to 
 -10 v. t, is determined primarily by the time for the plate of the gate tube 
 to reach a voltage (+73 v ) such that the left-hand flipflop grid reaches +1 v. 
 
 The equivalent circuit is: 
 
 / 
 
 +90 v, — 
 
 + 5?v. 
 
 
 ' T(5-T^L.~^'^>^'^ 
 
 8- 
 
 e" X = .284 
 
 = 1.26 = 
 
 RC 
 
 t x = (1.26)(5. D(8.10 
 
 t, = 5^ asps 
 
 t_ is determined by assuming the tube conducts a constant current 
 of 5 ma, 1 ma of which is used in the resistors, and determining the time re- 
 quired for E to reach -10 v. C = 15-9 M+if neglecting Miller effect. 
 
 O XO "Ccl-L 
 
 Vf- (1? f (10) = u 0m , s . 
 
 The time for the cathode follower to respond is: 
 
 cv _ (io.9)(io) _ 
 *3 ■" 1 " (15) " 7 * 3 
 
 mu.s 
 
 T, . , = t n + t_ + t„ = 101 mu-s 
 total 12 3 
 
 -8- 
 
Measurements : 
 
 The measured delay was 125 mus . This was measured in a racing register 
 where the tubes were neither turned on or off instantaneously as was assumed in 
 the calculations. However, the delays were long enough so that the equilibrium 
 conditions were reached. 
 
 Example IV. 
 
 ■Go 
 
 Symmetrical Pentode Flipflop 
 with "Grid " Gate Feeding Cathode Follower 
 
 -H5& 
 
 -hoo 
 
 ^SltCjU? 
 
 
 ■-^^2£51 i5 
 
 IN ®- 
 
 7t.^<o 
 
 ->r\0o 
 
 A 
 
 
 YtZ& 
 
 OUT o- 
 
 100^/of 
 
 '.2-8 tC 
 
 
 /oSe-wf: 
 
 ^2_6J6 
 
 -J- li/j^ 4. ^k- t "• e 
 
 £>AW& 
 
 4» fc> 
 
 I-IO 
 
 ^A^fc 
 
 6.S5K ~ 
 |w. 
 
 ' o / -hlOO 
 
 - !5~<? 
 
 /z2C5l 
 
 OOUT 
 
 
 I5*> 
 
 This circuit was used in a two stage racing register. The capacities 
 shown are measured values. The values in parentheses are the capacities which 
 can be assumed to exist from the point indicated to ground. 
 
 R L = k. 5 K 
 
 E, = +50 
 beq 
 
 -9- 
 
Calculations : 
 
 The current for pulling the plate down is I = 11 ma. The average 
 current used in the resistors is 1 ma. 
 
 The swing required to actuate the next gate is 5 v so: 
 
 CV. .126X5). 
 
 *i-i 
 
 av 
 
 10 
 
 = 13 mu.s 
 
 The time required for the cathode follower to respond to this change 
 
 is : 
 
 cv = iioK|l m 6>7 
 
 7.5 
 
 mu-s 
 
 The upgoing speed is largely determined "by the plate circuit of the 
 gate tube. The equivalent circuit is: 
 
 1 
 
 +60 
 
 — +zo 
 
 
 ^-a.^^-f 
 
 -YAO 
 
 -\2D 
 
 20 -x 
 
 U0 = e 
 
 = .50 
 
 x = 
 
 .69 = 
 
 RC 
 
 t 3 = (.69)(5.D(8.5) 
 
 t = 30 mu-s 
 
 T D + Ty = 13 + 6.7 + 30 = 49. 7 mus. 
 
 Measurements : 
 
 In the racing register the value for T + T was 100 mus. In this 
 case, as before, the tubes were not turned on and off instantaneously. 
 
 -10- 
 
Example V. 
 
 +?DO 
 
 Flipflop with Bi-Directional 
 Secondary Emission Tube Gate 
 
 + 3£o 
 
 This flipflop was driven by two 20 mu.s pulses separated by 100 mus . 
 The waveforms which were obtained are shown below: 
 
 All times in mu.s 
 
 Note that there was considerable "ringing" at the output. It was 
 not possible to stop this effect in the experimental circuit. 
 
 The calculations for this circuit are difficult since the current 
 
 in the EFP60's cannot be accurately estimated. A first order approximation 
 
 will be made: 
 
 From direct- current characteristics with E., = 150 v, E _ = 250 vm 
 
 dynode ' g2 
 
 = h0 ma. Then assuming 
 
 dynode 
 
 E n = -1 v. we obtain I — 5k ma, I 
 gl ' V 
 
 i = K(E _ - uB , ) ' and evaluating K and p. from the tube curves we find 
 P g2 gl & 
 
 -11- 
 
for E , = and E _ = 150 that I = ^5 ma and I, J = 33 ma. 
 gl g2 p dynode 
 
 So the time for the output point to go up 20 v is T 
 
 u 
 
 m CV (25)(20) lc 
 
 T = - = * 'L.1 — L = 15 mas 
 
 u !j ^ 33 
 
 dynode 
 
 The time to go down 20 v is T : 
 
 „, CV (25)(20) _. 
 
 T d = I" = "45 = U ms ' 
 
 P 
 
 Example VI, 
 
 Flipflop with Bi-Directional Gate 
 4r5"0 
 
 +?0 
 
 4-100 
 
 4 
 
 I IQ&* 
 
 2.7IC. 
 
 I.ZK 
 
 >4_ oj. 
 
 fe 6B<%7a 
 
 
 I^ h / 
 
 _L_Z8 c ^-f' 
 Owl) 
 
 X Jj\ 6J6 
 
 -HOO 
 1 
 
 J_ ^A <b3& 
 
 I.2K. -±- 
 
 0*) 
 
 +4 
 4-2,6 
 
 
 M 
 
 -V44 
 
 -Z| 
 
 ?o^4 
 
 ^1L £ 
 
 ISK. 
 
 4-1 OCJ 
 
 4 4 
 
 •H6 
 
 -* — * |i 
 
 
 6^6 
 
 -12- 
 
 1 
 
 /ok 
 
 
 -\00 OUT 
 
 -IOO 
 
 -\oo 
 
 -12- 
 
This circuit was used in a three stage racing register. The capacity 
 values are averages of those measured in the actual circuit. The voltages are 
 measured values. It is clear that the capacities cannot he taken as the actual 
 capacities from these points to ground since there is some capacitive coupling 
 from one point to another. The values in parentheses represent values which 
 can he considered to exist at the given points. 
 
 Calculations : 
 
 The average current for pulling the plate down is I = 13 ma. 
 The swing required "before the flipflop holds if 6 v so: 
 
 av 
 The time required for the cathode follower to respond to this change 
 
 is : 
 
 is 
 
 The time required for the flipflop cathode to respond to this change 
 
 v ..ffl.iiflga_^ ; 
 
 T, = t, + t + t = 29.6 mus 
 d 1 2 3 
 
 The upgoing speed is largely determined by the 2.7 K resistor in the 
 gating circuit. The plate voltage at which the flipflop holds is +38 v. 
 
 -13- 
 
The equivalent circuit is 
 
 <7dy. 
 
 2.7 K 
 
 -o* o ^A— 
 
 r 
 
 ^Zov, 
 
 \OjyJl 
 
 12 
 
 30 
 
 = e 
 
 -x 
 
 = .h 
 
 x = .915 = W 
 
 T = (.92)(2.7)(10) = 25 mus 
 
 T, + T = 29.6 + 25 = 5^-6 mus 
 d u 
 
 Measurements : 
 
 The measured time for T, + T = 60 mas. Note that the inclusion of 
 
 d u 
 
 t^ in the calculations nearly compensates for the fact that the tubes were 
 neither turned on or off instantaneously. This is a coincidence. 
 
 k. Construction Practice 
 
 The construction of high speed circuits requires the use of practices 
 similar to high frequency RF work in the several hundred MC region. Each supply 
 voltage line must be bypassed at every point to which connections are made. 
 The filament lines must be bypassed at almost every tube. 
 
 A piece of wire two inches long has an inductance of 0.06 u.hy and with 
 a shunt capacity of 10 [i\if will produce a resonant circuit at 200 Mc. Lead 
 lengths must be kept short therefore to prevent parasitic oscillations since 
 the tubes are still operative at this frequency. 
 
 5. Pulse Generators 
 
 In general less difficulty was encountered in obtaining very short pulses, 
 10-50 nps, than in obtaining circuits to use them. The circuits which worked 
 fastest were those using secondary emission tubes such as the EFP60. Free running 
 
 -14- 
 
as well as driven pulsers were constructed and pulses as short as 10 mu.s were 
 obtained. The most reliable circuit from the standpoint of reproducibility 
 was the following free running pulser: 
 
 EpPfcS 
 
 follows : 
 
 —Zoo 
 
 The length of the pulse was reduced by adding successive stages as 
 
 +300 
 
 -t-500 
 
 +\50 
 
 A 
 
 ZZ.O-TL- 
 
 I w. 
 
 4 
 
 • ■\_ 
 
 zzo_n_ 
 
 l_ 
 
 2.2.O. 
 
 A 
 
 70jy^ 
 
 
 4'ic 
 
 
 Z2.0_q_ 
 
 160^- 
 
 •»- OUTPVT 
 
 ~ZOO 
 
 -ZOO 
 
 -15- 
 
Waveforms 
 
 The essential design features of the 'divider' stage are that the 
 inductance L is chosen so that the second overshoot of the plate circuit 
 occurs after the grid pulse has ended and that the natural period of the 
 stage as a free running pulser is several times the period of the driver; in 
 the example above the feedback capacitor from plate to cathode is made twice 
 that of the driver stage so the ratio of periods is about two. 
 
 6. Conclusions 
 
 I feel that operation times less than 50 m\is in a racing register 
 type circuit are very difficult to obtain with present tubes and circuit 
 practices. Operation times of about 30 ups might be obtained by using fast 
 pulses and pulse driven circuits. With this latter method the gates can take 
 advantage of a duty cycle and the full charging currents for the shunt capa- 
 cities are available for the entire operation time. 
 
 Acknowledgement 
 
 J. M. Wier has worked on most of these circuits with the author. 
 The configurations are primarily due to Mr. Wier. 
 
 GHL/hc 
 
 -16- 
 
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 Research and Engineering 
 Swiss vale, Pennsylvania 
 Attn: L. 0. Grondahl, Director 
 
 General Electric Company 
 Director of Research 
 Schenectady 5> New York 
 Attn: Dr. Irving Langmuir 
 
 -5- 
 
Institute of Math. Sciences 
 AEC Computing Facility- 
 New York University 
 25 Waverly Place 
 New York, New York 
 
 University of Pennsylvania 
 
 Moore School of Electrical Engineering 
 
 Philadelphia k, Pennsylvania 
 
 Attn: Dr. M. Rubinoff 
 
 Columbia University 
 New York 27, New York 
 Attn: Prof. F. J. Murray 
 
 RCA Laboratories 
 Princeton, New Jersey 
 Attn: Dr. Jan Rajchman 
 
 University of Minnesota 
 Institute of Technology 
 Minneapolis lU, Minnesota 
 Attn: Dr. S. E. Wars chaws ki 
 
 David Taylor Model Basin 
 Navy Department 
 Carderock, Maryland 
 Attn: Dr. H. Polacheck 
 
 Remington Rand, Inc. 
 315 ^th Avenue 
 New York 10, New York 
 Attn: Dr. Grace Hopper 
 
 Chief, Mathematics Panel 
 
 Oak Ridge National Laboratory 
 
 Oak Ridge, Tennessee 
 
 Attn: Dr. A. S. Householder 
 
 -6-