224 Royal Society: — 



precisely similar to that for the addition of ordinary algebraical quan- 

 tities ; as regards their multiplication (or composition), there is the 

 peculiarity that matrices are not in general convertible ; it is never- 

 theless possible to form the powers (positive or negative, integral or 

 fractional) of a matrix, and thence to arrive at the notion of a 

 rational and integral function, or generally of any algebraical func- 

 tion of a matrix. I obtain the remarkable theorem that any matrix 

 whatever satisfies an algebraical equation of its own order, the coeffi- 

 cient of the highest power being unity, and those of the other powers 

 functions of the terms of the matrix, the last coefficient being in fact 

 the determinant. The rule for the formation of this equation may be 

 stated in the following condensed form, which will be intelligible 

 after a perusal of the memoir, viz. the determinant, formed out of 

 the matrix diminished by the matrix considered as a single quantity 

 involving the matrix unity, will be equal to zero. The theorem 

 shows that every rational and integral function (or indeed every 

 rational function) of a matrix may be considered as a rational and 

 integral function, the degree of which is at most equal to that of the 

 matrix, less unity ; it even shows that in a sense, the same is true 

 with respect to any algebraical function whatever of a matrix. One 

 of the applications of the theorem is the finding of the general ex- 

 pression of the matrices which are convertible with a given matrix. 

 The theory of rectangular matrices appears much less important 

 than that of square matrices, and I have not entered into it further 

 than by showing how some of the notions applicable to these may 

 be extended to rectangular matrices. 



January 21.— Dr. J. D. Hooker, V.P., in the Chair. 



The following communication was read : — 



" On the Physical Structure of the Old Red Sandstone of the 

 County of Waterford, considered with relation to Cleavage, Joint 

 Surfaces, and Faults." By the Rev. Samuel Haughton, Fellow of 

 Trinity College, Dublin, and Professor of Geology. 



After describing the general features of the district and giving his 

 reasons for selecting it, the author proceeds to give a detailed account 

 of the faults, joint surfaces, and cleavage planes, 345 in number, 

 observed by him during the course of his survey. 



The faults are nineteen in number and reducible to two pairs of 

 rectangular systems. The bearings of these systems are E. 7° 30' N., 

 and E. 34° 22' N. The other faults, which form nearly right angles 

 with the preceding and may be called Conjugate Faults, have the 

 following bearings, N. 3° 45' W. and N. 33° 24' W. 



The author considers that the existence of two systems of conju- 

 gate faults indicates two distinct systems of upheaving force in the 

 district ; a supposition which is strongly confirmed by the fact that 

 the average strike of the beds is E. 10° 46' N., a direction inter- 

 mediate between those of the systems of faults. He then demon- 

 strate s from 345 observed planes, that the systems of joint and 

 cleavage planes are also conjugate systems, reducible to four, of which 

 two are identical with the two conjugate systems of faults already 



