Notices respecting New Books. 227 



As a further illustration of our meaning we will consider a pro- 

 perty of plane curves, which is not noticed in the usual text-books, 

 and which M. Euchonnet attributes to M. Abel Transon (p. 42) j 

 viz. that if 6 is the angle between the normal and the diametral 

 curve at the same point of a given curve, 



tan 0=*.* 

 3 ds 



Let y =f(x) be the equation to the curve, the point under considera- 

 tion (O) being taken as the origin, and the axis of x so chosen as 

 to touch the curve ; consequently /(0) and /'(0) are severally zero. 

 Suppose the curve to be cut by a chord PP' parallel to Ox ; if A 

 is its middle point, the ultimate value of AOy is d. If (h, Tc) and 

 (—h',7c) are the coordinates of P and P', it is plain that tan is 

 the ultimate value of (h—h')+2Jc. Now, observing that it is not 

 necessary to retain terms above the third order, we have 



k=f(h) =if"(0).h»+}.f»(0).h\ 

 and k=f(-h)=if"(0) . h*--L ./'"(0) . h>\ 



Hence ^fHf^l, 



and A»«JL J ! + *.«.*}. 



/»(0)l ^ d /"(0) J 



Therefore, subtracting and dividing out h-\-h', the ultimate value 



°f o, is — ^ . r44rp. But it follows, from the well-known general 



f"(0) 

 expression for the radius of curvature, that — rzffTTyi > 2 is the value 



P_at the origin ; and this proves the theorem. 

 as 



Elements de Ccdcul Ajoproanmatif. Par Chaeles Euchonnet (de 

 Lausanne). Seconde edition augmentee. Paris : Gauthier-Vil- 

 lars. Lausanne : Georges Bridel. Zurich : Orell, Puessli et 

 Comp. 1874 (8vo, pp. 65). 



The question discussed in the work before us is this : — What 

 precautions must be taken in a numerical operation to ensure that 

 the first n digits of the final result shall be exact ? The author con- 

 siders separately the operations of addition (and subtraction), multi- 

 plication, division, extraction of roots, and, very briefly, the case 

 of a function of one variable. He illustrates his rule by working 

 out several examples ; but he does not insert examples for practice, 

 as an English writer would probably have done. The above-men* 

 tioned operations become complicated when any or all the numbers 

 concerned are incommensurable ; and in these cases a second ques- 

 tion arises, viz. to what degree of approximation these numbers 

 must be taken separately to ensure the required degree of accuracy 

 in the final result. 



We shall perhaps convey the best idea of the book by descri- 



R2 



