TRANSACTIONS OF THE SECTIONS. " 27 



coefficients are equal^ or there is at least one of them greater than and om l^ss than 



Subdivide the interval X— a;;, iuto ti part.? : the -^, corresponding to one of them, 

 .?•(— .r,_i, will be greater than ,^j — ^ (No. II.). Operate iu the same manner with 

 .r,— .ivi, and so on. We shall thus hare an indefinitely increasing series of — ^'s,all 

 greater than ^~'^° (No. IT.), and having for limit the difierential coefficient y' of 

 Fx for a certain value of x (No. I.). There is, then, a differential coefficient y, 

 greater than ~^°. In the same way we can show that there is one smaller. We 

 must, however, except the case of ^ constant, which aiises when y=ax+b. 



rV. If the differential coefficient of a function, supposed the same in the positive 

 andtiegative directiotis, is equal to a constant a, from .r„ to X, the function is linear 

 and of the form ax-\-b. 



Necessarily, x and .x\ being any two values included in the interval (^x^, X), 



y-.Vi_ 



a, 

 1 



') 



whatever .r and x^ may be. For, were it otherwise, there would be between .v and 

 Xj a differential coefficient greater than a, and one smaller than a. Thus :— 



y=ax+(y^—ax^). 

 Corollary.— li a=0, y = constant. Q.E.D. 



On Convergerds. By Thomas Mttie, M.A., F.B.S.E. 



In Lagrange's additions to Euler's Algebra (2nd Eng. ed. vol. ii. p. 279), he sets 

 himself the problem,— ^//-ac^io?/ expressed by a great number of figures being given, 

 to find all the fractions, in less terms, which ajjproach so near the truth that it is im- 

 possible to approach nearer without employing greater ones ; and for solution he gives 

 in effect the following rule -.—Transform the given fraction into a continued fraction 

 with unit numerators and positive integral paiiial denominators, and the so-called con- 

 fer gents of this continued fraction ivill be the fractions required. In this he is in error, 

 the' fi-actions found being some of the fractions required, but not all. Thus, taking 

 IT as the given fractional form, he transforms it into 



111 



3+ 



o 



7+ 15+ 1+. 



the so-called convergents of which are j, f , 'j^^, 'jy^, 5 and i^i I'^gai'd to them 



he says :— " So that we may be assured that the fraction \ approaches nearer the 

 truth than any other fraction whose denominator is less than 7 ; also the fraction 

 y approaches nearer the tnith than any other fraction whose denominator is less 

 than 106 ; and so of others." 



The statement here made in reference to j is easily seen to be incorrect by com- 

 paring the difference of \ fi-om tt with that of ", % or ^, the former being, of 



course, -14159 . . . , and the three latter -10840 . . . , -05840 . . . , '02507 . . . ; and 

 the incon-ectness extends to what is said of the other convergents. The ti-ue solu- 

 tion lies in the fact that not only is 3 ■\-\ one of the required fractions, but so also 



a* 



