The curvature of the distended belly at the navel is found to be, from the foregoing measurements,

Multiplying this curvature into the tension of the abdominal muscles at the navel already found, viz. 133.67 lbs. per inch, we obtain, finally,

P= 133.67=32.926 lbs. per square inch.

4.0596

This amount of expulsive force per square inch is available, although not usually employed, to assist the uterus in completing the second stage of labour. If we suppose it applied to the surface of a circle 4 inches in diameter (the usual width of the pelvic canal), we find that it is equivalent to 523.65 lbs. pressure.

Adding together the combined forces of the voluntary and involuntary muscles, we find

Involuntary muscles

Voluntary muscles

Total

= 54.106 lbs.

=523.65 ""

577.75 99

Thus we see that, on an emergency, somewhat more than a quarter of a ton pressure can be brought to bear upon a refractory child that refuses to come into the world in the usual manner*.

In order to determine by actual experiment the expulsive force of the abdominal muscles, I placed two men, of 48 and 21 years of age respectively, lying on a table upon their backs, and put a disk measuring 1.87 inch diameter just over the navel; weights were placed upon this disk and gradually increased until the extreme limit of weight that could be lifted with safety was reached; this limit was found to be in both cases 113 lbs. As the circle whose diameter is 1.87 inch has an area of 2.937 square

* The preceding result will no doubt remind the curious and well-informed reader of the statement made by Mr. Shandy, on the authority of Lithopædus Senonensis, De partu difficili,' that the force of the woman's efforts in strong labour pains is equal upon an average to the weight of 470 lbs. avoirdupois acting perpendicularly upon the vertex of the head of the child.

inches, the pressure perpendicular to the abdominal wall produced by the action of the abdominal muscles was

a result which differs little from that already found by calculation from the actual measurements of the muscles and curvatures.

II. "Tables of the Numerical Values of the Sine-integral, Cosineintegral, and Exponential Integral." By J. W. L. GLAISHER, Trinity College, Cambridge. Communicated by Professor CAYLEY, LL.D. Received February 10, 1870.

called the sine-integral, cosine-integral, and exponential integral, were used by Schlömilch to express the values of several more complicated integrals, and denoted by him thus,-Six, Cix, Eix; the last function, however, is for all real values of a only another form of the logarithm-integral, the relation being

Ei x=lie*.

These functions have since been shown to be the key to a very large class of definite integrals, and several hundreds have been evaluated in terms of them by Schlömilch, De Haan, &c., so that for some time they have been considered primary functions of the integral calculus, and forms reduced to dependence on them have been regarded as known.

Considering, therefore, the large number of integrals dependent on them for their evaluation, and their consequent importance as a means of extending the integral calculus, it seemed very desirable that they should be systematically tabulated, the only values which have previously been obtained being those of Six, Cix, Ei x, Ei (-a) for the values =1, 2, ... 10 calcu lated by Bretschneider, and printed in the third volume of Grunert's Archiv der Mathematik und Physik,' and a Table of the logarithm-integral published by Soldner at Munich in 1806.

The present Tables contain the values of Six, Cix, Eix, Ei (-x) for values of a from 0 to 1 at intervals of '01 to nineteen places of decimals, for values of a from 1 to 5 at intervals of 1, and from 5 to 15 at intervals of unity, to ten places, and for a=20 to twelve places. Also values of Six and Cix only for values of a from 20 to 100 at intervals of 5, to 200 at intervals of 10, to 1000 at intervals of 100, and for several higher values to seven places; besides Tables of the maxima and minima values of these functions, corresponding in the case of the sine-integral to multiples of ", and in

π

the case of the cosine-integral to odd multiples of, also to seven places.

III. "Researches on Solar Physics.-No. II. The Positions and Areas of the Spots observed at Kew during the years 1864-66, also the Spotted Area of the Sun's visible disk from the commencement of 1832 up to May 1868." By WARREN DE LA RUE, Esq., Ph.D., F.R.S., F.R.A.S., BALFOUR STEWART, Esq., LL.D., F.R.S., F.R.A.S., Superintendent of the Kew Observatory, and BENJAMIN LOEWY, Esq., F.R.A.S. Received February 15, 1870.

(Abstract.)

The paper commences with a continuation for the years 1864-66 of Tables II. and III. of a previous paper by the same authors; it then proceeds to a discussion of the value of the pictures of the sun made by Hofrath Schwabe, which had been placed at the disposal of the authors, and the result is that these pictures, when compared with simultaneous pictures taken by Carrington and by the Kew heliograph, are found to be of great trustworthiness. From 1832 to 1854 the pictures discussed are those of Schwabe, who was the only observer between these dates; then follows the series taken by Carrington, and lastly the Kew series, which began in 1862.

A list is given of the values of the sun's spotted area for every fortnight, from the beginning of 1832 up to May 1868, and also a list of threemonthly values of the same, each three-monthly value being the mean of the three fortnightly values which precede and of the three which follow it. These three-monthly values are also given for every fortnight.

A plate is appended to the paper, in which a curve is laid down representing the progress of solar disturbance as derived from the three-monthly values; and another curve is derived from this by a simple process of equalization, representing the progress of the ten-yearly period. The values of the latter curve, corresponding to every fortnight, are also tabulated. From this Table are derived the following epochs of maxima and minima of the longer period :

This exhibits a variability in the length of the whole period.

Thus we have between 1st and 2nd minimum......

9.81 years.

Another fact previously noted by Sir J. Herschel is brought to light,

namely, that the time between a minimum and the next maximum is less than that from the maximum to the next minimum.

Thus the times from the minimum to the maximum are for the three periods 3.06, 4.14, and 3.37, while those from the maximum to the minimum are 6-75, 8:44, and 7.44 years.

In all the three periods there are times of secondary maxima after the first maximum; and in order to exhibit this peculiarity, statistics are given of the light-curve of R Sagittæ and of ẞ Lyræ, two variable stars which present peculiarities similar to the sun.

Finally, the results are tested to see whether they exhibit any trace of planetary influence; and for this purpose the conjunctions of Jupiter and Venus, of Venus and Mercury, of Jupiter and Mercury, as well as the varying distances of Mercury alone in its elliptical orbit, have been made use of, and the united effect is exhibited in the following Table, the unit of spotted area being one-millionth of the sun's visible hemisphere :

IV. "On the Contact of Conics with Surfaces."

By WILLIAM SPOTTISWOODE, M.A., F.R.S. Received February 16, 1870.

(Abstract.)

It is well known that at every point of a surface two tangents, called principal tangents, may be drawn having three-pointic contact with the surface, i. e. having an intimacy exceeding by one degree that generally enjoyed by a straight line and a surface. The object of the present paper is to establish the corresponding theorem respecting tangent conics, viz. that "at every point of a surface ten conics may be drawn having sixpointic contact with the surface;" these may be called Principal Tangent Conics. In this investigation I have adopted a method analogous to that employed in my paper "On the Sextactic Points of a Plane Curve" (Phil.

Trans. vol. clv. p. 653); and as I there, in the case of three variables, introduced a set of three arbitrary constants in order to comprise a group of expressions in a single formula, so here, in the case of four variables, I introduce with the same view two sets of four arbitrary constants. If these constants be represented by a, 6, y, d, a', B', y', ', I consider the conic of five-pointic contact of a section of the surface made by the plane w-kw=0, where w=ar+By+yz+dt, and w'=a'x+ẞ'y+y'z+d't, and k is indeterminate; and then proceed to determine k, and thereby the azimuth of the plane about the line w = 0, '=0, so that the contact may be sixpointic. The formulæ thence arising turn out to be strictly analogous to those belonging to the case of three variables, except that the arbitrary quantities cannot in general be divided out from the final expression. In fact, it is the presence of these quantities which enables us to determine the position of the plane of section, and the equation whereby this is effected proves to be of the degree 10 in w: '=k, and besides this of the degree 12n-27 in the coordinates x, y, z, t (n being the degree of the surface), giving rise to the theorem above stated.

Beyond the question of the principal tangents, it has been shown by Clebsch and Salmon that on every surface U a curve may be drawn, at every point of which one of the principal tangents will have a fourpointic contact. And if n be the degree of U, that of the surface S intersecting U in the curve in question will be 11n-24. Further, it has been shown that at a finite number of points the contact will be five-pointic. The number of these points has not yet been completely determined; but Clebsch has shown (Crelle, vol. lviii. p. 93) that it does not exceed n(11n—24) (14n-30). Similarly it appears that on every surface a curve may be drawn, at every point of which one of the principal tangent conics has a seven-pointic contact, and that at a finite number of points the contact will become eight-pointic. But into the discussion of these latter problems I do not propose to enter in the present communication.

March 17, 1870.

Capt. RICHARDS, R.N., Vice-President, in the Chair.

The following communications were read:

I. "On the Law which regulates the Relative Magnitude of the Areas of the four Orifices of the Heart." By HERBERT DAVIES, M.D., F.R.C.P., Senior Physician to the London Hospital, and formerly Fellow of Queens' College, Cambridge. Communicated. by W. H. FLOWER, Hunterian Professor of Comparative Anatomy. Received January 27, 1870.

I propose in this communication to inquire whether any law can be discovered which determines the relative magnitude of the areas of the