led me to investigate the results of the congenital coarctations of the aorta at or beyond its isthmus. I have been interested to find that in a large percentage of these cases of coarcted aorta there is dilation beyond the site of the coarctation. The generally accepted view that this dilation is to enable the aorta to better carry on the anastomotic circulation must, it seems to me, be erroneous. When we shall have ascertained the cause of the arterial dilation obtained experimentally below constricting bands and of the dilation of the artery proximal to an arterio-venous fistula we may be able to explain the dilation of the aorta beyond the congenital coarctation.

C. Plausible explanation of the presence of blood in lymph-cysts at the second and subsequent tappings.-A few years ago, assisted by Dr. Heuer, I removed from the abdomen of a woman about forty years of age a huge congenital hygroma or lymph-cyst. The diaphragm was pushed high up into the right thorax and the liver was displaced far to the right and so rotated on its vertical axis that its inferior border instead of being transverse was parallel and almost in line with the linea alba. The enucleation of the greater part of the cyst was easily accomplished, the few adhesions being disposed of by gentle, blunt dissection. Finally, when there remained only a few filamentous fibers binding the sac to the right adrenal gland1 and the inferior vena cava we proceeded with even more deliberation and caution. The adhesions to the vein were so delicate that the gentlest manipulation with the handle of the scalpel sufficed to break them. We had an unusually free and clear exposure of the vein and were operating without embarrassment. Suddenly blood gushed from a linear defect about 3 mm. long in the vena cava. The hemorrhage was promptly controlled and the slit in the vessel.sutured. Proceeding thereafter with perhaps even greater delicacy, we were again confronted with a gush of blood from the vena cava at a higher point. Here we found a slit from 1.5 to 2 cm. long in this vein. The edges of the slit were smooth, the linear defect being clearly not due to a tear or cut. The gap in the vein was closed by suture. Dr. Heuer and I satisfactorily assured ourselves that there was no defect or special thinning of the wall of the cyst at the point contraposed to the larger of the two defects in the wall of the vena cava.2

The slits in the wall of the vena cava were surely not artefacts. They represented, I believe, imperfectly closed embryonic communications between the vein and lymph buds or lymphatic vessels. Dr. Florence Sabin to whom we owe so much for our knowledge of the origin and development of the lymphatic system very kindly writes me in regard to this case as follows:

Recent work on the lymphatic system serves to demonstrate that lymphatic vessels are modified veins. It has been shown that lymphatic vessels occur first in the neck as sacs, lined with endothelium and packed with blood, which lie close to the jugular veins. The abdominal lymphatics begin as a sac which lies close to that part of the inferior cava which connects the two Wolffian bodies. Baetjer showed in 1908 that in the pig this sac which is the forerunner of the retroperitoneal lymphatics communicates for a time with the inferior vena cava. These communications between the lymphatics and the abdominal veins which are transitory in the pig were then shown to be permanent in the South American monkeys

by Silvester in 1912 (lantern slides), while in 1915 Job demonstrated similar permanent connections in rodents. Thus the study of the development of the lymphatic system affords an explanation of anomalies involving connections between the lymphatic vessels and both the renal veins and the inferior vena cava.

The statement has repeatedly been made that hygromata which at the first tapping have yielded a clear fluid may be found at all subsequent tappings to contain more or less blood. Only one explanation has been offered for the presence of the blood, viz., trauma of the wall of the cyst. This explanation has always seemed to me an unsatisfactory one because the walls of these cysts are as a rule very thin and non-vascular. May it not, in view of the findings in our case, be possible that vestigia of lymphatico-venous communications are responsible for the admixture of blood which has occasionally been noted at only the second and subsequent tappings of lymph-cysts and is more frequently found at the first tapping? The negative pressure consequent upon the aspiration of fluid from the cyst might divert for the moment a little blood from the vein which had given origin to the hygroma's lymphatic bud or vessel. Thus the contents of the sac, clear at the first withdrawal, would be blood stained at the second. Thereafter, with each tapping blood would be aspirated into the sac and hence clear fluid might never again be obtained.

The relation of the cyst to the right adrenal gland was remarkable. In the course of stripping the sac's final delicate attachments we exposed a flat, black surface, evidently the spread-out medulla of the adrenal, about the size of a half dollar. Parenchymatous oozing from this surface required for its arrest a few mattress sutures of fine silk.

2 The patient recovered promptly and has enjoyed excellent health since the operation.

TABLES OF THE ZONAL SPHERICAL HARMONIC OF THE SECOND KIND Q1(z) AND Qı′(z)

BY A. G. WEBSTER AND WILLARD FISHER

BALLISTIC INSTITUTE, CLARK UNIVERSITY*

Communicated, January 9, 1919

In connection with certain investigations that I was asked to make with regard to submarines it seemed convenient to have tables of the zonal harmonic of the second kind. These are desirable in various problems in mathematical physics connected with ellipsoids of revolution, such as problems in magnetism. Failing to find such tables in the literature of the subject, I requested Dr. Fisher to calculate tables of the function of the first order and its derivative for an interval that will be sufficient for most practical problems. Dr. Fisher has now left the country, and the stress of other work that prevented their publication being over, I trust that they may be available for peaceful purposes.-A. G. W.

* Contribution from the Ballistic Institute, Clark University, No. 1.

1.14 .5543 .5519

.5495

1.10 0.6745 0.6708 0.6671 0.6635 0.6600 0.6565 0.6530 0.6495 0.6461 0.6428 1.11 .6394 .6362 .6329 .6297 .6265 .6234 .6203 .6172 .6141 .6111 1.12 .6081 .6052 .6023 .5994 .5965 1.13 .5799 .5773 .5746 .5720 .5694 .5471 .5448

1.15 .5310
1.16 .5096 .5075

.5288 .5266

.5244

.5222

.5179 .5158

.5937 .5909 .5881 .5854 .5826
.5668 .5643 .5618 .5593 .5568
.5424
.5200

.5401 .5378 .5355 .5332

.5137 .5116

0.0286 0.0270 0.0255 0.0241 0.0228 .0170 .0162 .0155 .0148 .0142

3.0 0.0397 0.0370 0.0346 0.0324 0.0304 .0217.0206 .0196 .0186 .0178 .0137 .0131 .0126 0121 .0117 .0113 .0108 .0104 .0101 .0098

4.0

5.0

6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 0.0094 0.0080 0.0069 0.0060 0.0052 0.0047 0.0042 0.0037 0.0034

Below the line ****, second differences are negligible. Prefix the sign

to every value

1.10 3.7158 3.6710 3.6270 3.5839 3.5417 3.5004 3.4598 3.4201 3.3811 3.3429 1.113.3054 3.2687 3.2326 3.1972 3.1624 ||3.1283 3.0948 3.0619 3.0296 2.9979 1.12 2.9667 2.9360 2.9059 2.8764 2.8473 2.8187 2.7906 2.7629 2.7358 2.7090

1.132.6827 2.6569 2.6314 2.6063 2.5817 2.5574 2.5335 2.5100 2.4869 2.4641 1.14 .4416 .4195 .3977 .3763 .3551 .3343 .3138 .2935 .2736 .2540 1.15 .2346 .2155 .1967 .1781 .1598 .1418 .1240 .1064 .0891 .0720 1.16 .0551 .0385 | .0221 .0059 1.9899 1.9741 1.9586 1.9432 1.9280 1.9130 1.17 1.8982 1.8836 1.8692 1.8550 .8409 .8270 .8133 .7998 .7864 .7731 1.18 .7601 .7472 .7344 .7218 .7093 .6970 .6849 .6728 .6609 .6492 1.19 .6376 .6261 .6147 .6035 .5924 .5814 .5706 .5598 .5492 .5387

.3012

.2252

.3093 1.23 .2623 .2547 .2472 .2398 .2325 1.24 .1898 .1829 .1760 .1693 .1626 1.25 .1236 .1173 .1111 .1049 .0988 1.26 .0631 .0573 .0516 .0459 .0403 1.27 .0076 .0023 0.9970 0.9918 0.9866 1.28 0.9564 0.9516.9467 .9419 .9371 1.29 .9093 .9047 .9003 .8958 .8914

1.201.5283 1.5180 1.5079 1.4978 1.48791.4870 1.4683 1.4587 1.4491 1.4397 1.21 .4304 .4211 .4120 .4030 .3940 .3851 .3764 .3677 .3589 .3506 1.22 .3421 .3338 .3255 .3174 .2933 .2854 .2776 .2699 .2180 .2108 .2037 .1967 .1559 .1494 .1428 .1364 .1300 .0927 .0867 .0807 .0748 .0689 .0347 .0292 .0237 .0183 .0129 0.9815 0.9764 0.9713 0.9663 0.9614 .9324 .9377 .9230 .9184 .9138 .8870 .8827 .8784 .8741 .8698

.3454

.3444

.3362 .3352

.3342

.3263

.3254

.3244

.3169 .3160

.3150

.3069 .3061

1.50 0.3953 0.3940 0.3927 0.3915 0.3902 0.3890 0.3877 0.3865 0.3852 0.3840 1.51 .3828 .3816 .3804 .3791 .3779 .3768 .3756 .3744 .3732 .3720 1.52 .3709 .3697 .3685 .3674 .3662 .3651 .3640 .3628 .3617 .3606 1.53 .3595 .3584 .3573 .3562 .3551 .3540 .3529 .3518 .3507 .3497 1.54 .3486 .3475 .3465 .3413 .3402 .3392 1.55 .3382 .3372 .3312 .3302 .3292 1.56.3283 .3273 1.57 .3187 .3178 1.58 .3096 .3087 .3078 1.59 .3009 .3000 .2992 .2983 .2975 .2966

.3433

.3423

.3332

.3322

1.60 0.2925 0.2917 0.2908 0.2900 0.2892 0.2884 0.2876 0.2868 0.2860 0.2852 1.61 .2844 .2836 .2828 .2821 .2813 .2805 .2797 .2790 .2782 .2774

1.6 0.2925 0.2844 0.2767 0.2693 0.2621 0.2552 0.2486 0.2423 0.2361 0.2302 1.7 .2245 .2190 .2137 .2086 .2036 .1988 .1942 .1898 .1854 .1812 1.8 .1772 .1733 .1695 .1658 .1622 .1588 .1554 .1521 .1490 .1459 1.9 .1429 .1400 .1372 .1345 .1318 .1293 .1267 .1243 .1174 .1152 .1130 .1069 .1050 .1031 .0897 .0882 .0867

.1219 .1196

.1013.0995

.0853 .0839

0.0255 0.0253 0.0250 0.0247 0.0245 0.0242 0.0240 0.0237 0.0235 0.0232

3.0 0.0284 0.0255 0.0230 0.0208 0.0189 0.0172 0.0157 0.0144 0.0132 0.0122 4.0 .0113 .0104 .0096 .0090 .0083 .0078 .0073 .0068 .0064 .0060 5.0 .0056 .0053 .0050 .0047 .0044 .0042 .0039 .0037 .0035 .0034

6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 0.0032 0.0025 0.0020 0.0016 0.0013 0.0011 0.0009 0.0008 0.0007

The above tables were computed to five places with Barlow's Tables of squares and Naperian logarithms, checked by differences and the last figure. cut off. When the figure cut off was exactly 5, the number was recomputed to six places or seven places.