TRANSACTIONS OF THE SECTIONS. S3 



its whiteness, and by waving and irregular ridges and sulci on its surface. 

 The younger the subject, the thicker is this epithelium. It is most promi- 

 nent over the molar teeth ; its internal surface is concave, receiving the pro- 

 jecting mucous membrane; it is constituted by a mass of scales, super- 

 imposed on each other ; and hence the terms " dental cartilage," or cartilage 

 of the gum, which have hitherto been applied to this structure, give an 

 erroneous idea of its true nature ; for cartilage always presents the corpuscle 

 discovered and described by Purkinje. Here, as in other portions of the 

 epithelium, (except on the surface of the vascular mucous membrane, which 

 presents simple cells with their corpuscles,) the external scales are the 

 largest. In the interior of this alveolar epithelium, where it corresponds to 

 the molar teeth, small vesicles may be frequently observed, varying in size 

 from a quarter to an eighth of a line in diameter. To the naked eye they 

 are transparent ; under the microscope their parietes are found to consist of 

 attenuated scales, and their cavity to contain a fluid abounding in minute 

 granules or cells. The internal surface of the alveolar epithelium frequently 

 presents concavities, from a line and a half to three or four lines in circum- 

 ference, corresponding to projections from the mucous membrane, formed by 

 a larger species of vesicle, which is deeply implanted in the vascular mucous 

 membrane. The parietes of these vesicles are delicate membranes; they 

 contain a transparent fluid, coagulable by heat, acid, or spirit, in which float 

 numerous globules and scales similar to those of the epithelium generally. 

 The external surface of the alveolar epithelium presents, after a slight mace- 

 ration, numerous fringed processes, measuring I'rom one line to one line and 

 a half in length, and half a line in breadth. Under the microscope these 

 fringes are found to be composed of elongated scales connected together, 

 forming masses, which divide and subdivide, until they attain such an extreme 

 tenuity, that the most minute terminations consist but of two scales, in mar- 

 ginal apposition. If the epithelium be carefully separated from the surface 

 of the mucous membrane corresponding to the unextruded molar teeth, and 

 placed in water, or in diluted spii-it of wine for some little time, its external 

 surface presents these fringes much enlarged, and forming a mass more con- 

 siderable in size than the dense epithelium itself. 



MATHEMATICS AND PHYSICS. 



On the relation of Sturm s Auxiliary Functions to the roots of an Algebraic 

 Equation. By Professor Sylvester. 



The author availed himself of the present meeting of the British Association to 

 bring under the more general notice of mathematicians his discovery., made in the 

 year 1839, of the real nature and constitution of the auxiUary functions (so-called) 

 which Sturm makes use of in locating the roots of an equation : these are obtained 

 by proceeding with the left hand side of the equation and its first differential co- 

 efficient as if it were our object to obtain their greatest common factor ; the success- 

 ive remainders, with their signs alternately changed and preserved, constitute the 

 functions in question. Each of these may be put under the form of a fraction, the 

 denominator of which is a perfect square, or in fact the product of many : hkewise 

 the numerator contains a huge heap of factors of a similar form. 



These therefore, as well as the denominator, since they cannot influence the series 

 of signs, may be rejected ; and furthermore we may, if we please, again make every 

 other function, beginning from the last but one, change its sign, if we consent to 

 use changes wherever Sturm speaks of continuations of sign, and vice versa. 



The functions of Sturm, thus modified and purged of irrelevancy, the author, by 

 way of distinction, and still to attribute honour where it is really most due, proposes 



