Notices respecting New Books. 67 



nite or any similar igneous rock having been erupted at the sur- 

 face in the manner apparently taken for granted by M. Delesse, and 

 that rocks so erupted are always lava (whether doleritic or trachytic) 

 or basaltic rocks, such as those which he says have alone the marks 

 of decided igneous action. We think it most likely, then, that the 

 difference in the effects of the lavas and the granitic rocks is due, not 

 to the former having been more purely igneous than the latter, but 

 to the different conditions under which they have acted. 



Lessons Introductory to the Modern Higher Algebra. By the Rev. 

 George Salmon, A.M., Fellotv and Tutor, Trinity College, Dublin. 

 Dublin : Hodges, Smith & Co., 1859. 



Within the last eighteen years the old and well-trodden field of 

 Algebra has been invaded by a host of new and strange intruders, 

 with the odd-sounding names of ' Determinants,' ' Hyperdetermi- 

 nants,' ' Discriminants, ' Emanants,' ' Invariants,' ' Evectants,' 

 ' Bezoutiants,' ' Hessians ' (having no connexion, however, with 

 either 'Boots' or 'Crucibles'), ' Canonizants ' (of no religion), 

 ' Dialytics/ and ' Quantics.' Many a reader of the Cambridge 

 Mathematical Journal, the Philosophical Magazine, Philosophical 

 Transactions, &c., has wondered what it all meant — wondered 

 sometimes, indeed, whether there was any meaning at all in these 

 new expressions and their symbols. Very few even of the best 

 mathematicians of the day have paid much attention to the subject 

 as yet ; but they are beginning to do so, finding that there is really 

 something like a new branch growing out of their old tree — nay, 

 more, that this young off-shoot is already bearing fruit. This ' Alge- 

 bra of Linear Transformations' " may be said to date from a paper 

 published by Prof. Boole in the Cambridge Mathematical Journal for 

 Nov. 1841, which was mainly occupied with applications of the fol- 

 lowing theorem : — Let an ordinary algebraic equation be made homo- 

 geneous by writing x : y for x ; let these variables then be linearly 

 transformed by writing \x+piy, \'x + i.i'y for a; and y ; and let V=0 

 be the condition that the transformed equation shall have equal 

 roots ; then the function V will be equal to the similar function for 

 the original equation multiplied by a function of X, fx, X', y! , and not 



involving the coefficients of the original equation 



Mr. Boole having in the paper just cited made important use of the 

 principle here enunciated, Mr. Cayley subsequently proposed to 

 himself the problem to determine a priori what functions of the 

 coeflficients of a given equation possess this property of invariance ; 

 viz. that when the equation is linearly transformed, the same func- 

 tion of the new coefficients shall be equal to the given function 

 multiplied by a quantity independent of these coefficients. The 

 result of Mr. Cayley's investigations was to discover that the pro- 

 perty of invariance was not peculiar to the functions which had 

 been discussed by Mr. Boole, and to bring to light other important 

 functions possessing the same property. Subsequently it was found 

 that functions could be formed involving the variables as well as the 

 coefficients, and possessing the same permanent relation to the 



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