and the greatest height h to which the tree can grow without flexure is given by

where c is the least positive root of the equation

J_4m-1 (x) = 0.

n-4m+2

By assigning different values to m and n according to the growth of the tree, and to E, μ, and λ according to the elasticity and density of the wood, the greatest straight vertical growth of a tree can be inferred. The application of these formula to the case of the large trees of California would be interesting, but in the absence of the numerical data required, I am unable to carry this

out.

This paper was written for Dr Asa Gray, Professor of Botany in Harvard University, Cambridge, Mass., and was to have been read at the meeting of the American Association last year, but arrived too late.

A pine tree, as described in Sproat's British Columbia (1875), is said to have grown in one straight tapering stem to a height of 221 feet, and to have measured only 20 inches in diameter at the base.

Considered as an example of Article II., and neglecting the weight of the branches, a diameter of 20 inches at the base would allow of a vertical stable growth of about 300 feet.

Perhaps the best assumptions to make for our purpose as to the growth of a tree are, (i) to assume a uniformly tapering trunk as a central column, and (ii) to adopt Ruskin's assumption (Modern Painters), that the sectional area of the branches of a tree, made by any horizontal plane, is constant.

This is equivalent to putting m=1 and n = 1 in equation (2), and then the solution depends on the least root of the associated Bessel's function of the order - 3.

Generally, for a homogeneous body, n = 2m+1, and the diameter at the base must increase as the power of the height, which accounts for the slender proportions of young trees, compared with the stunted appearance of very large trees.

February 21, 1881.

PROFESSOR NEWTON, PRESIDENT, IN THE CHAIR.

T. H. Corry, Gonville and Caius College, F. R. Weldon, St John's College, and W. Heap, were balloted for and duly elected associates of the Society.

The following communication was made to the Society:

On the estimation of ferment in gland-cells by means of osmic acid. By J. N. LANGLEY, M.A., Fellow of Trinity College.

A few years ago Nussbaum1 observed that such ferments as he could obtain stained rapidly with osmic acid. In consequence of this observation he made many others upon the staining power with osmic acid of ferment-producing glands. He arrived at the conclusion that the depth of staining with osmic acid was a satisfactory indication of the ferment-content of gland-cells.

Mr Langley briefly reviewed the facts that have been brought forward for and against Nussbaum's method and conclusions. He contended:

That the cells of different glands do not increase in staining power in proportion to the amount of ferment that can be obtained from them.

That the cells of any one gland do not increase and decrease in staining power as the cells increase and decrease in various physiological states in ferment-content.

That there is no obvious correspondence between the depth of staining with osmic acid of extracts of ferment-producing glands and the amount of ferment contained by the extracts. This point was illustrated by experiments.

From these facts it follows that the depth of staining with osmic acid of any gland-cell is not a satisfactory indication of the power of the cell to produce ferment.

There are very few physiological substances which, when isolated, reduce osmic acid readily. Those that have this action, such as bile acids, hæmoglobin and peptone, do not, as far as we know, occur in living gland-cells; this taken together with the fact that glandular tissues diminish in staining power when they

1 Nussbaum, Archiv f. Mik. Anat. Bd. xIII. S. 746, 1877.

2 Nussbaum, Archiv f. Mik. Anat. Bd. xv. S. 119, 1878; Bd. xvI. S. 543, 1879. Edinger, Archiv f. Mik. Anat. Bd. xvII. S. 193, 1879.

3 Grützner, Pflüger's Archiv, Bd. xvi. S. 122, 1877; Bd. xx. S. 395, 1879. Langley, Foster's Jour. Physiol. Vol. 1. p. 271, 1879.

are kept for some hours before being put in osmic acid suggests that the staining of the normal tissue depends upon the presence of some unstable constituent of the living protoplasm not at present isolated.

March 7, 1881.

PROFESSOR NEWTON, PRESIDENT, IN THE CHAIR.

The following communications were made to the Society: (1) On the Action of the Vagus Nerve upon the frog's heart. BY W. H. GASKELL, M.D., Trinity College.

The chief object of this communication was to demonstrate by means of curves certain effects of vagus stimulation as obtained by a new method.

The method used was the following:

The heart was cut out, with vagus attached, the bulbus aortæ held tightly by means of Kronecker's forceps, the ventricle slit open and the extreme apex attached by a thread to an ordinary lever, so that the strip of cardiac tissue between the bulbus and the apex alone moved the lever and recorded its movements on the blackened drum.

Also simultaneous tracings of the beats of the auricles and ventricle and of the base and apex of the ventricle were obtained by using an upper and lower lever and clamping, not too tightly, the heart, either between the auricles and ventricle or midway between the base and apex of the ventricle. The curves obtained by these methods demonstrated the following facts:

1. Weak stimulation of the vagus causes simply a marked increase in the force of the contractions of the ventricle with, as a rule, a slight acceleration in the rate of beat.

2. Stronger stimulation causes a diminution in the force of the contractions followed by an increase; both during and after the stimulation the rate of rhythm is frequently increased.

3. When strong stimulation causes a complete standstill the ventricular contractions increase in force and rapidity beyond the normal after the stimulation has ended.

4. The stimulation of the nerve may cause steady increase in the force of the auricle contractions with simultaneous diminution even to complete standstill of the ventricle contractions. After the stimulation the increase in the ventricle contractions occurs when the auricular have already begun to diminish.

5. When the ventricle is beating alternately weakly and strongly, stimulation of the nerve makes all the beats equally strong.

6. This alternation of beats may occur in the apex of the ventricle alone while the base beats regularly, and then the vagus stimulation causes the apex beats to become equally strong.

Further by the method used the action of poisons, such as Atropin, Muscarin, Digitalin, on different parts of the heart and on the effect of the vagus nerve was demonstrated, and it was shewn that:

1. When the beats of the heart were reduced in size as by the application of normal saline solution, then Atropin Sulphate 1 p. c. solution applied to the heart caused an increase in the force of the contractions with a slower rhythm.

2. Atropin applied to the heart caused a slower rhythm with strong contractions and removed all the various effects of vagus stimulation.

3. Atropin applied to the ventricle only did not prevent the action of the nerve on the ventricle.

4. Atropin applied to the auricles and sinus only removed the whole effect of the nerve from all parts of the heart.

5. Muscarin applied to the ventricle only reduced the size of its contractions and relaxed the tissue without altering the rhythm, at all events at first, and without affecting the auricles in the least.

6. When after the application of Muscarin or Digitalin the ventricle had ceased beating in a relaxed or semi-contracted condition respectively, then Atropin in each case brought back the beats without altering the condition of the tissue.

The author suggested from the consideration of these and many other curves which he possessed, that a possible explanation of the action of the vagus might be found on the hypothesis that the vagus is the trophic nerve of the cardiac muscle. He however could not at that stage of his investigation give any definite explanation of the phenomena observed by him, but trusted that further experiments would enable him to do so.

(2) On the ancestral form of the Chordata. By F. M. BALFOUR, M.A., F.R.S., Fellow of Trinity College.

March 21, 1881.

PROFESSOR NEWTON, PRESIDENT, IN THE CHAIR.

Dr Armistead, Dr G. M. Bacon, Mr H. Baxter, Mr R. Bowes, Mr Bumpstead, Mr J. Carter, Mr A. Deck, Mr H. Gotobed, Mr A. Graham, Mr Marshall (Ely), Mr W. E. Pain, Mr A. Schuster, and Mr W. W. Smith, were balloted for and re-elected associates of the Society.

The President stated that it was with very great pleasure that he was enabled to place upon the table the charter of the Society, which had been out of their possession since 1852; and he was glad to think that the remarks made by him at the Annual Meeting on October 25 had led to its recovery. They were much indebted to Professor Paget and Messrs. S. and W. Peed for the assistance they had rendered, and the Council had passed a cordial vote of thanks to them for their services.

The following communications were made to the Society:

(1) On Conjugate Functions of Cartesians and other Quartics. By A. G. GREENHILL, M.A.

1. The well-known case of the conjugate functions of confocal ellipses and hyperbolas in which x+iy is a trigonometrical function of+in may be considered as derived from the integral

and the foci of the system are at z=0 and z = 1.

or

and

If r, r' denote the distances of a point from the foci, then