4 REPORT 1865. 



cone, in such a way that a theorem relating to one figure frequently establishes a 

 corresponding theorem relating- to the other. Thus many properties of conies in 

 general arc at once suggested and proved by reference to the circle. 



Again, the theory of reciprocal polars, or rather the principle of duality, which 

 enables us to see points and straight lines in a condition of interdependence such 

 that theorems relating to points (e. g. positions or curves, intersections of lines, &c.) 

 at once give rise to corresponding theorems relating to straight lines (tangents, rec- 

 tilinear loci, &c). Under the head of modern geometrical methods falls also the 

 theory of pencils of rays and transversals ; straight lines radiating from a point and 

 cutting another line, straight or curved. This again suggests the idea ot relations 

 between the segments of the transversal (when straight) or between the angles 

 made inter se by the radiating lines. The most fruitful conception of this kind lias 

 been that of the auharmonic ratio of four points or rays. This peculiar ratio re- 

 mains unchanged under such a variety of circumstances, that it has arisen to an 

 almost independent principle in geometry ; and upon it M. Chasles may be said to 

 have founded, to a very great extent, his ' Geometric Superieure ' and his new 

 work on conic sections, the first volume of which has recently appeared. Before 

 quitting this part of the subject, it should not be omitted that a great part of these 

 theories have their application to figures in space as well as to those in piano. 



The second volume of M. Chaales's work will contain a full exposition of his recent 

 most important contribution to the theory of conies. lie has found that the properties 

 of a system of conies satisfying any four conditions whatever may be most naturally 

 expressed in terms of two elements or characteristics — namely, the number of such 

 conies which pass through any point, and the number which touch any line. 

 Starting from this fruitful notion, he has, by a process which may be termed geo- 

 metrical substitution, been able to express, in a single symmetrical formula, the 

 number of conies which satisfy any five conditions whatever. We may almost say 

 that he has condensed into this formula the whole theory of conies. 



Again, connected with this is the principle of deformation — another method of 

 considering one figure in relation to another, the points of the one being connected 

 by a definite construction with those of the other. By this, and in particular by a 

 most happy extension of it by Professor Hirst, theorems and properties of curves of 

 higher degrees are demonstrated through those of lower, e.g. curves of the fourth 

 and fifth degrees by conies. 



Passing to analysis, we have in the first place the analogues of the geometrical 

 theories above mentioned. To the method of projection corresponds (in one of its 

 interpretations at least) the method of linear transformation ; to that of deformation, 

 non-linear transformations. The method of transversals as well as those of auhar- 

 monic ratio and geometrical involution admit of a concise analytical statement ; 

 but they cannot be called methods even in analytical geometry, still less in analysis 

 proper. The principle of duality, however, as treated by Pliicker, may claim an 

 analytical with as good a right as a geometrical basis. 



Before quitting this part of the subject, mention should be made of two impor- 

 tant and original contributions to analytical geometry in space. One, by Professor 

 ( Jayley, is directed to the representation of curves in space ( by means of cones having 

 variable vertices), a method free from the extraneous branches sometimes intro- 

 duced by the ordinary conception of the complete intersection of two surfaces ; of 

 the other, by Professor Pliicker, we have at present only the abstract in the pro- 

 ceedings of the Royal Society; it promises, however, to abound in processes of 

 great power and originality. 



But the greatest acquisition to modern analysis is what is now generally termed 

 the new algebra. This calculus, which originated in this country, and from the 

 first received wide developments at the hand of its founders, Boole, Cayley, and 

 Sylvester, has, during the last few years, found numerous cultivators both amongst 

 ourselves and on the Continent. The main problem proposed for solution is the 

 investigation of tin 1 properties of rational homogeneous algebraical functions of any 

 number of variables, the forms to which they are capable of being reduced, the 

 subsidiary expressions to which they give rise or with which they may be asso- 

 ciated, and the bearing of the latter upon the former. Investigations so general, 

 so abstract, and so apparently removed from any practical application could not 



