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XXVI. Notices respecting New Books, 



Exposition Geometrique des Proprietes Generates des Courbes. Par 

 CHA.ELES RucHONtfET (de Lausanne). Troisieme edition, aug- 

 mentee et en partie refondue. Paris : Grauthier-Villars. Lau- 

 sanne: G-eorges Bridel. Zurich: Orell, Fuessli et Comp. 1874 

 (8vo, pp. 160). 



rpHIS work contains a very complete account of the properties 

 -*• of curved lines which may be considered general, as distin- 

 guished from those possessed by certain curves at individual or 

 singular points. Though the author has occasion to notice that 

 there may be points of inflection, he does not consider them, as 

 inflection cannot be a property enjoyed by the points of a curve 

 generally. Thus the proof of art. 44, that in general a curve cuts 

 its osculating plane at the point of contact, consists in showing 

 that if the curve do not cut the osculating plane at an assigned 

 point, there must be inflection at the corresponding point of another 

 curve. 



The consequence of this is, that the book consists mainly of a 

 discussion of curvature, and of the properties of the osculating 

 circle and of allied subjects. Thus the five sections which make 

 up the first part of the work (that devoted to plane curves) treat 

 of the tangent, curvature, the osculating circle, and expressions" 

 for different magnitudes : viz. (1 ) those which arise out of a con- 

 sideration of an infinitely small arc and the tangents at its extremi- 

 ties, such as that the difference between the arc and its chord 

 ultimately equals one 24th of the square of the angle of contin- 

 gence multiplied by the length of the arc ; (2) those which depend 

 on the difference between the radii of curvature at two infinitely 

 near points of the curve, such as that the difference between the 

 angles contained by the chord and the tangents at its extremities 

 is ultimately equal to one half of the square of the angle of con- 



tingence multiplied by -£• 



In the second part (that devoted to tortuous curves, and which 

 occupies about three quarters of the volume) the subjects are much 

 the same ; but, of course, many more points come under notice ; 

 and, besides, there is an account of Ruled and Developable Surfaces, 

 sufficient to enable the reader to understand the properties of the 

 polar and rectifying surfaces of these curved lines. 



Our author's treatment of the subject as a whole is both minute 

 and exact; and though we do not profess to be acquainted with the 

 whole literature of the subject, it is only fair to say that we do 

 not know any work which contains an account of it to be com- 

 pared with that before us in completeness. The chief peculiarity, 

 however, of the treatment is in its method, which is wholly that 

 of limits — each proposition being separately proved by reasoning 

 directly from a diagram, without the use of any of the elementary 

 formulae of the differential calculus. Thus nothing so recondite 



Phil. Mag. S. 4. Vol. 49. No. 324. Maech 1875. R 



