1870.] On the Areas of the four orifices of the Heart. 265 



Trans, vol. civ. p. 653) ; and as I there, in the case of three variables, in- 

 troduced 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 in- 

 troduce with the same view two sets of four arbitrary constants. If these 

 constants be represented by a, ft, y, d, a', ft', y, d\ I consider the conic 

 of five-pointic contact of a section of the surface made by the plane 

 7ff—/cru'=0, where "ni = a,x+fty + yz + ht s and ?z' = a!x + ft'y-\-y'z-\-h't, and 

 k\s indeterminate; and then proceed to determine Jc, and thereby the azimuth 

 of the plane about the line zzr = 0, tjj'—O, so that the contact may be six- 

 pointic. The formulae 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 w T hereby this is 

 effected proves to be of the degree 10 in w : zj'=/c, and besides this of the 

 degree I2n— 27 in the coordinates cc, y, s 3 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 four- 

 pointic contact. And if n be the degree of U, that of the surface S inter- 

 secting U in the curve in question will be \ \n— 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(l In— 24) (I4n — 30). Similarly it appears that on every surface a 

 curve may be drawn, at every point of which one of the principal tangent 

 conies 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. 33 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 Ana- 

 tomy. 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 



