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cal values of the coefficients A and B. There is no change in algebraic sign and the range in their values is so small that the simultaneous determination of x and y becomes practically indeterminate. For the middle zone the conditions are more favorable, but even here it has not seemed advisable to attempt a direct solution. Since the quantity x, as shown by the uniform-field solution, is apparently constant throughout the entire interval, we have preferred to solve by successive approximations, beginning with an assumed value of x.

We therefore write the fundamental equation in the form

B tan i cos


Ax' - A,



where for convenience x' k/cos i has been substituted for 1/x. The value of x' from the uniform-field solution is 1.00, and this we use for the first approximation in deriving Y = tan i cos λ from (3). We have for each zone, from all the data for each day, the normal equation

[BB] tan i cos


[BA] x' [AB]


The individual values of Y are presumably in error, because of the assumed value for x'. It is easily shown, however, that neither the phase nor the period of the curve Y = tan i cos λ is thereby affected. The entire effect goes into the amplitude. Using the first approximation for x', we can therefore determine to and P for each zone, free from error, exactly as in the case of the uniform-field solution. The curves for Y show that the results in each case are sensibly the same as those previously found, and we therefore adopt for each zone the values given above.

Since the weight of tan i cos λ given by (4) is [BB] the normal equation for tan i is

Σ [AB] cos X

tan i Σ [BB] cos2 X = x' Σ [BA] cos A (5) where the outer summation symbol covers all the separate days. With this equation we find for the three values of i

I 4.9 ± 0.7; II 6.7 ± 0.5;

III 3.7±1°0


With the aid of these values we now determine from the measures of each day a new approximation for x', using the normal equation

[AA] x = [AA] + [BA] tan i cos


which is at once derived from the fundamental equation. Noting that the weight of each x' is [AA] and combining the different days, we have for the mean x'

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It thus appears that for zones I and III the original assumption was nearly correct; for the second zone, however, the change in x' is large.

For the further approximations we have from (5) and (8), respectively,

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For zones I and III, a and ẞ are small quantities of the order of 0.01 or 0.02, and since dx', the differences between the first and second approximations for x', are only -0.03 and -0.04, respectively, equations (10) and (11) show that the corresponding values of i and x' given in (6) and (9) are final.



For zone II, a = 0.120 and 8 = 1.42. We have found already, in order, x'1 = 1.00, tan i = 0.118, x'2= 0.60, whence dx' = -0.40; and by successive substitution into (10) and (11) we derive tan i2 0.070, x'3 = 0.53, tan is 0.062, x4 = 0.52, tan i 0.061. As no further change is produced by additional substitutions, x' and tan is are adopted as the final values. Calculating k from x' and collecting results, we have



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10 S to 45 S,

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0.28 days

to 1914, June 25.38 ± 0.42 days, G. M. T.

The uncertainty in the inclinations averages about 0.7, while that in the k's is one or two units of the second decimal. Since the inclinations, periods, and epochs for the three zones are sensibly the same, the deviations from a uniform spherical field within the region from 45° N. to 45° S. are apparently restricted to the polar field-strength, which is inversely proportional to k. Here the variation with latitude is unexpectedly large-far greater than the uncertainty of the calculation. Unless the observations are affected by a large systematic error, which is itself a rather complicated function of the magnitude of the observed displacement A, the change must be real.

The measures have all been made with the parallel-plate micrometer and it is difficult to believe that errors of the kind required to make the result illusory can have entered. Our experience would lead us to believe that important variations in the field-strength of the character indicated actually exist.

It is unexpected to find the i's for all the zones smaller than that from the uniform-field solution. It is easily seen, however, that the two series of results stand in the proper relation to each other. In the uniform-field solution the displacements within the limits of zones I and III contribute about fifteen


times as much to the weight of k as do those in zone II. The mean result for k should therefore be sensibly the same as the values for zones I and III when treated separately, as proves to be the case (k 0.99, 96, and 96, respectively). The weight of the inclination, on the other hand, is derived largely from the observations of the equatorial zone. Moreover, i is calculated after k has been found, so that the conditions in the uniform-field solution are analogous to those of the first approximation of zone II. In this approximation the equatorial observations combined with x'=1.00—the equivalent of k = 0.99—gave i = 6.7, which is nearly the same as the value found from all the data. The measures of zones I and III combined with k = 0.99, as we know from the zonal analysis, must lead to relatively small values for i, amounting to about 4°. The mean therefore lies between these two extremes, with a preponderance of weight in zone II. This accounts for the value of 6.0 originally found—a mean result which is in excess of all the inclinations found by treating the zones separately. It is only when the equatorial observations are discussed by themselves that the true value of k for this region reveals itself or has any appreciable influence upon the solution; but when once found the inclination is necessarily decreased.

We are under great obligation to Miss Wolfe of the Computing Division who has rendered much assistance with the extensive numerical calculations required for the discussion of the data.

1 Seares, F. H., van Maanen, A., and Ellerman, F., these PROCEEDINGS 4, 1918, (4–9).




Read before the Academy, April 28, 1919

In eight experiments in a series of thirteen, after giving meat in large quantities (700 to 1300 grams) to a dog weighing 11 kgm., the respiratory quotients during the height of protein metabolism were between 0.793 and 0.800. A calculation showed that under these conditions the retained carbon residue of the protein metabolized was held back in such a form that, had it been oxidized, the respiratory quotient of this retained pabulum would have been 0.85. This would represent the oxidation of material half of whose calories were composed of fat and half of carbohydrate. The dog showed quotients of 0.82 and above only after the larger quantities of meat were given (1000 grams or more). It was extremely difficult to induce the dog to take meat in sufficient quantity to indicate a considerable production of fat from protein. Incidentally it was observed that the basal metabolism of a dog fed with meat in large quantity for a time and then caused to revert to a 'standard diet'

(meat, 100 grams; lard, 20 grams; and biscuit meal, 100 grams) remained persistently (even after 2 weeks) at a higher level than had obtained prior to the meat ingestion. This confirms F. G. Benedict's idea of a higher basal metabolism in the presence of 'surplus' cellular nitrogen, or the 'improvement quota' of protein according to Rubner's suggestive terminology. When meat was given after the partial depletion of the body cells of their 'improvement quota', protein was retained in greater measure, less protein was metabolized and the heat production was therefore lower than on a subsequent day (experiments 46, 47; 51, 54).

The following table gives the method of calculation followed:

Experiment 55. N in urine per hour = 1.58 grams

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= 0.797

R. Q. of deposit = 0.83

R. Q. per hour-0.81, 0.74, 0.80, 0.84. R. Q. for whole period

In the above table:

N-CO2 is the amount of CO2 derivable from the protein metabolism during an hour.

N-O2 is the amount of O2 necessary to oxidize the protein metabolized in one hour.

Resp. CO2 and Resp. O2 are the amounts of CO2 and O2 which were actually. respired during an hour.

The difference represents (1) the CO2 which would have been expired had all the retained carbon of the protein metabolism been oxidized and (2) the O2 which would have been employed in that process. The relation between the volumes of these two gases indicates that the material retained and unoxidized would have yielded a respiratory quotient of 0.83, which indicates the retention of a pabulum, approximately half of the calories of which were derived from fat and half from glucose.

N-cals. is the quantity of heat which would have been eliminated by the dog had all the protein metabolized by the dog been completely oxidized. From this is subtracted the number of calories estimated to have been retained as a mixture of fat and glucose aforesaid. The difference represents the calories as calculated by indirect calorimetry, which in this case agrees exactly with those directly measured by the calorimeter by the method of direct calorimetry.

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* Standard diet at 5 p.m. and thereafter daily until March 15.




Communicated by E. H. Moore, April 29, 1919

Note II on Conformal Mapping under aid of Grant No. 207 from the Bache Fund.

Let w = w(z) be a power series in z, convergent for |z|<1 and such that the circle <1 is mapped conformally on a simple (that is, simply connected and nowhere overlapping) region in the w-plane. By a linear transformation w1 = aw+b, we may reduce w(z) to the form z + a2z2 + . . . + anz” +

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