a transformation of y in terms of z of the second order. From the simultaneous values x = 0, y = √(− ic), corresponding to a = K, it follows that

Conversely these two equations may be proved to be consistent, and to lead to the differential relation

I am indebted to Captain P. A. Mac Mahon, R.A., for the discovery of a misprint in Cayley's Elliptic Functions, which retarded the verification; and also to Mr Pilkington, Fellow of Pembroke College, for the algebraical verification of this theorem. The misprint 1-i k is on p. 74, and consists in putting cn (K+iK')=

-(-ic), instead of (-ic), as it should be.

or

√n, where n is of the form 4p-1; or

x = cn α, y = cn } (1 + i √n) a,

w=K-iK'

;

The form of the transformation may be inferred from the considerations that

(1) when 1 - y = 0,

} (1+i√n) a = 2mK+2im'K' (m+m' even),

(m + im2√/n) (i − i √n) K,

} (1 + i √n) a = (2m + 1) K + (2m' + 1) iK' (m + m' even),

} (1 + i √n) a = (2m + 1) K+ (2nı' + 1) iK' (m + m' odd), a = (4m' + 2) K + (2s + 1) w,

y+ic = RII (cn 2sw + x)

+ D,

y-ic = SII {cn (28 + 1) w + x} ÷ D;

and P, Q, R, S are determined from the simultaneous values = 1 y=1 and x = 0, y=√(— ic).

By a repeated application of the process we can express

cn {}(1+i√n)}'a

in terms of cn a by a transformation of the order p.

The case of n = 11 presents peculiar difficulties; we therefore

K'

K

pass on to the case of =√15, then (kk') = sin 18° (Joubert, Comptes Rendus, t. 50);

where a = cn (K – ¿K'), ß = cn ‡ (K–¿K'); and therefore aß=ic; also

These results have also been verified algebraically by Mr Pilkington,

It is not possible to express en (1+i√5) a rationally in terms of cn a; but if x = cn a, y = cn (1+i√5) a, then y can be expressed in terms of x by the sextic transformation

K'

We had

=

K

= √√3, and a, wa, w'a the parameters of the elliptic

functions of the third kind in the reduction of the integral

and it will be an interesting subject of enquiry to determine the simplest form of the integrals which reduce to elliptic integrals of the third kind in which

K'

=

Κ

:√√/ɔ̃, √7, √15,...

and the parameters involve the corresponding complex multipliers.

(2) On certain points in the function of the cardiac muscle. By W. H. GASKELL, M.D., F.R.S.

PART I. On the rhythmical properties of the cardiac muscle.

IN a former paper1 I have described experiments which brought me to the conclusion that the beat of the frog's heart was dependent on separate impulses proceeding from the sinus to the auricles and ventricle and travelling along distinct nerve paths. Also, the modifications of the sequence of ventricular upon auricular beat caused by various methods of procedure, such as tightening the clamp between auricle and ventricle, heating the auricles and sinus, the application of various poisons &c., were all discussed on the supposition that a contraction can only occur when a certain relation exists between the strength of the impulse and the excitability of the muscular tissue.

In a subsequent paper I have shown by experiments upon the heart of the tortoise that the conception of separate impulses passing from the sinus to the auricles and ventricle is unnecessary, and that all the variations of sequence observed in the case of the frog as well as those noticed in the tortoise, are to be explained by the more or less complete blocking of a contraction-wave, which starting at the sinus passes over the auricles and when it reaches the ventricle causes a contraction there. I still however imagined that the mediation of some nervous apparatus such as ganglion cells was involved in the transference of the contraction from auricle to ventricle.

Finally in a paper read before the British Medical Association last August, I showed that no need exists for the intervention of

1 Phil. Trans. 1882.

2 Journ. Physiol. Vol. I. p. 369. 3 British Med. Journ. 1882, p. 572.

special ganglionic apparatus to conduct the contraction from sinus to auricle or from auricle to ventricle, for in both the frog and the tortoise a band of circularly disposed muscular fibres exists at the sino-auricular and auriculo-ventricular junctions, through which the contraction wave is able to pass from sinus to auricle and from auricle to ventricle respectively.

Further the resemblance between the artificial blocking of the contraction wave by means of section to the natural block at the sino-auricular and auriculo-ventricular junctions was pointed out, and it was shown that the pauses between the contractions of the several cavities of the heart could be ascribed to the nature of the muscular tissue at these junctions together with the anatomical arrangement of the fibres there.

The ultimate conclusions therefore obtained from those parts of these three papers which dealt with the beat of the heart are the following:

1. The beats of the heart represent a series of separate peristaltic contractions which start from the sinus end and pass in regular order over the whole heart.

2. The peristaltic nature of these contractions is obscured by the fact that the wave of contraction passes along a muscular tube which is not of the same diameter or structure throughout; so that the contractions of certain portions, which by bulging have become more prominent and by a higher development of structure have become more rapid in their contraction, are so pronounced as to give the impression of separate contractions of these portions with pauses between them.

3. The conduction of this wave of contraction can be artificially hindered by various methods in any part of its course; so that if the hindrance is but slight an artificial pause is produced, if greater a partial block, in which case every second contraction passes the block, if still greater an increasing block until at last the block is complete and no contraction-wave is able to pass.

So far I have but slightly touched upon one most important point in this conception of the heart's rhythm, viz. the nature of the starting point of this contraction-wave. It is the consideration of this point to which I desire to call attention in this part of the present paper.

In the frog the septum between the two auricles plays an important rôle in all discussions upon the nature of the rhythmical contractions which occur after the removal of the sinus. In the tortoise on the other hand the septum possesses no such importance, its place is taken by a band of tissue, which passing from the sinus to the ventricle connects the two auricles together and is the path along which the coronary veins and the nerve trunks