53 



He also observes (p. 39) that when A 05 , B&, O, &c. are the high- 

 est powers of A, B, C, &c. which a polynome, agreeing in other 

 respects with [s] M , contains, the terms of such incomplete polynome 

 will agree in point of number with those terms in [s] M which are 

 not divisible by either A a + 1 , or B^+ 1 , or 0+ 1 , &c. The polynomes 

 from which Bezout proposes to eliminate certain terms, contain 

 terms of all dimensions from to u inclusive. The terms which are 

 to remain after the others have been eliminated, and which are enu- 

 merated by means of the condition, that they are not divisible by 

 certain powers of A, or B, or C, &c, may be of all dimensions indis- 

 criminately from to u inclusive. Bezout's object is exclusively 

 Elimination, and he makes no allusion to any other application of his 

 formulae. 



The polynomes considered by the author, taken in their entirety, 

 agree in their general structure with those considered by Bezout ; 

 but the nature of the author's inquiries led him to confine his atten- 

 tion to the composition of those particular terms in a polynome 

 which were of the same dimension ; and to seek to express the 

 number of the terms, not of all dimensions indiscriminately, but of 

 each particular dimension separately. To show how it has hap- 

 pened that researches, very different at their point of departure, 

 have, as regards one point of investigation, ended in nearly similar 

 formulae, the author proceeds to deduce his formula (VIII*.) from 

 the investigations of Bezout. Such a deduction, he conceives, might 

 readily have been made by any one to whom it had occurred to make 

 it ; and the application of such a deduction, when once made, to pro- 

 blems in Combinations, would have been much too obvious to have 

 remained long unnoticed. 



Expressions of the form above considered are regarded by Bezout 

 as of the nature of Differences ; and the truth of this view of the 

 subject may be shown in the following brief manner. 



If <p(x) generates $(u), [ 1 -x«] <p(x) will generate ^(m) — \J/(w — a), 

 which we may denote by A a rJ/(w). Consequently [1 — x&~\ [1 — #*] 

 <p(x), that is to say, 



{ 1 Tf'+< W }-#« 



will generate A 2 A>(u) ; and so on ; the independent variable, u, 

 undergoing, not uniform, but variable decrements, a, (5, y, &c. 



May 3, 1847. 



On the Internal Pressure to which Rock Masses may be subjected, 

 and its possible influence in the Production of the Laminated Struc- 

 ture. By W. Hopkins, M.A., F.R.S. 



If a plane of indefinitely small extent pass through any proposed 

 point in the interior of a continuous solid mass in a state of con- 



