February 2, 1899] 



NA TURE 



3'9 



iti-iil reduclion of Andrews' measurements at two points, 

 namely, the different temperatures of the manometer and the 

 level dift'erence q. The former might be supplied by making 

 fresh accurate investigations, but will the latter always remain 

 something to be longed for? 



The same remark seems to apply to the published results of 

 experiments by Janssen on nitrous oxide, and by Roth on car- 

 bonic acid, aethylen, sulphurous acid, and ammonia 



Imperial University, Tokj'o, Japan. K. Tsuruta. 



TiiF, letter above was sent to me some time ago with a re- 

 quest that I should forward it to Nature after making in- 

 quiries as to the possibility of obtaining, from the laboratory 

 books of Dr. Andrews, the desiderata which Prof Tsuruta 

 points out. 



I am delighted to find that one so well qualified is ready to 

 undertake the labour of the necessary reductions ; and I will 

 prepare for publication the data required for the purpose. A 

 recent inspection of the Note-books has shown me that they 

 contain the complete details of the experimental part of Dr. 

 Aiiilrews' great investigation. 



The work is by no means one of mere transcription ; it 

 requires great care, and therefore cannot be done in a hurry. 



Edinburgh, January iS. P. G. Tail 



Fourier's Series. 



The difficulty referred to by Prof. Michelson in Nature of 

 ( Jctober 6, 1898, and in subsequent letters, to that in your 

 issue of January 19, involves a disregard of the distinction which 

 it is necessary to make between a quantity which, however 

 small it may at first be taken, is thereafter to be kept fixed, and 

 a quantity which can be or is absolutely zero. To this distinction 

 there is the analogous one between a quantity which is arbitrarily 

 large but still considered as limited, and a quantity which is 

 entirely unbounded. 



The question considered by Prof Michelson, interesting as it 

 is, whether the limit, when n increases indefinitely, of the 

 quantity 



/(«, ;/) = sin 6 -i- J sin 2e 4- ... 4- - sin Kf, 



wherein € = /-ir/ji {k fixed and < 2h), is kit. is not really 

 pertinent as a criticism of the usual statement that the sum of 

 the series 



f (x) = sin.v -I- J sin 2.T -F ... -I- - sin nx -f ... to c<= 



is A(ir - .v) when o < .r < 2ir and is o when .r = o ; to get the 

 sum of such a series it is always to be understood (i.) that we 

 first settle for what value of -v we desire the sum, (ii.) that we 

 then put the value of ,v in the series, (iii.) that we then sum the 

 first II terms and find the limit of this sum when n increases in- 

 definitely, keeping x all the time at the value settled upon. In 

 the function f(f, n] above, this condition is not observed ; as 

 JI increases indefinitely, e := kirlii does not remain fixed, but 

 diminishes without limit. A similar convention is to be observed 

 in other cases. For instance, when a function of .x' is defined 

 by a definite integral taken in regard to a variable /, the variable 

 .V entering as a parameter in the subject of integration ; the 

 value of the function is then always to be found under the 

 hypothesis of a specified value for ,1 , which is to be substituted 

 in the subject of integration before the integration in regard to 

 e is carried out. Or, again, in such a common operation as 

 finding the differential coefficient ; for instance, we have 



— f.t-'^ cosf*] = 2 .r cos («■■')+ sin( ^-^ I . e-' 



which is indeterminate when .1=0; but the differential co- 



efficient of f {x\ = .X- cos( e'^ j at .v = o, is not indeterminate 



'L"„[/| 



o+/i]-f{o 



]A = ^[^-(^0]^°- 



for we have 

 li: 



.\nother point involved is that a function may continually 

 strive to a limit and yet not reach it. For instance, consider 

 ,. . 1 



.^1 \ « , limit /, -\ 

 « — 00 \ / 



NO. 1527, VOL. 59] 



where it is understood that we are to obtain the value of (/>(.v) 

 for any specified values of x by first substituting this value of x 



on the right side, then calculating the successive values of I - x" 

 for successive finite large values of n, and noticing the limit 

 towards which these values approach indefinitely. "When x has 

 any small specified and fixed value, however small, it follows, 



since the limiting value of - is o, that i^(.v) = .r- ; but when 



.1 = o, ,v" = o, and <^(-v) = I. The function thus continually 

 strives to the value o as .r approaches o ; but it does not reach 

 this value (see .also Gauss, "Werke," iii. p. 10). Unless I 

 mistake Mr. Hayward's letter of January 19, there is a similar 

 point there involved. The point P (in the sixth line of his 



letter from the bottom) strives to the point (ir,-J; in the sense 



in which the sum of a Fourier series is understood, it does not 

 reach this point. 



The discontinuity of the sum of the Fourier series considered 

 above is explained by the fact that as .v is taken near zero the 

 convergence becomes indefinitely slow ; the sequence of values 

 of ;/ necessary to make s - s,t of assigned smallness has infinity 

 for (an unreached) upper limit. 



I should be glad to take this opportunity of referring to a 

 point intimately connected with the considerations above, in 

 regard to which most of the accounts in the text-books appear 

 capable of more definiteness. The condition' that a sequence 

 of finite qualities S|, .t.,, ..., s„, .f„+i, ... should tend to a limit 

 is that for any specified small e it be possible to find a finite 

 m, such that for n > in and for all values of p the absolute 

 value ol s„+p - s„ should be less than €. The question may be 

 asked : Does all values of p mean only all finite values however 

 great (arbitrarily or indefinitely but not infinitely great), or is 

 the value/ = c« supposed to be required. There is no doubt 

 the phrase may be limited to mean all finite values of p, however 

 great. Thus taking the function (/>(.<) above, and putting, what 



is in accordance with the condition as now stated, s^ = <p\-\, 



the sequence of quantities <p ( - j defines a value, namely zero, 



which we may quite fairly describe as the limit of the sequence. 

 Though we may also say, in a certain sense, that this limit is not 

 reached ; in fact, the value s^^ regarded as ^(o), is i. And, 

 further, the series 



u^ I- u.. 4- «3 -I- ... 

 wherein 



;/j = I, u„ = 



:.-)-H^) 



is convergent, and its sum is the limit of the sequence j,, j.j, ..., 

 s„, ..., namely zero ; and this notwithstanding that j-^ = 1. A 

 more striking case is got by replacing i^(.v) by 



,i\ 1 , I'm /. -;:\ 



,f,{x) = X- +^ I - .r " , 



« = CX3 \ J 



The phraseology is analogous to the usual one for a definite 

 integral ; for instance, the integral 



J .r[logx-']'+.^' 



wherein .v is less than I, and a- is positive, has a definite limit 

 when C = o ; for whatever assigned value e may have, it is 

 always possible to find a positive value for (, such that for any 

 positive value of Ci less than f the integral 



f< dx 



J ..[logx- ]■+- 



is numerically less than f. This statement is, however, made 

 with the proviso that („ is not to be taken zero or infinitely 

 near to zero, though it may be taken as small as we please ; it 

 is indefinitely small without being infinitely small. If S,',) were 

 taken zero, the last integral would, strictly, be meaningless. 

 If only for the purpose of showing that the notion of an 



1 See, for instance, die excellent book of Harkness and Morley, " In- 

 troduction to Analytic Functions " (January iSgg; Macmillan and Co.), 

 which is surely unequalled for the matters of which it treats. 



