4i8 



NATURE 



{March i, 1888 



either case only appears in the first power ; its square 

 cannot be a divisor of the cyclotome. 



It is easy to prove the important theorem that no two 

 cyclotomes to the same base can have any the same 

 external divisor.^ 



We thus arrive at a result of great importance for the 

 investigation into the existence or otherwise of perfect 

 odd numbers, which (it being borne in mind that in this 

 theorem the divisors of a number include the number 

 itself, but not unity) may be expressed as follows : — 



The sum of a geo7netrical series whose first term is unity 

 and common ratio any positive or negative integer other 

 than -\- I or I — must contain at least as many distinct 

 prime divisors as the number of its terms co7itains divisors 

 of all kinds ; except ivhen the commojt ratio is - 2 or 2, 

 and the nwnber of terms is even in the first case, and 6 or 

 a multiple of 6 in the other, in which cases the nufnber of 

 prime divisors may be one less than in the general case.- 



In the theory of odd perfect numbers, the fact that, in 

 every geometrical series which has to be considered, the 

 common ratio (which is an .element of the supposed 

 perfect number) is necessarily odd prevents the exceptional 

 case from ever arising. 



The estabhshment of these laws concerning the divisors 

 and mutual relations of cyclotomes, so far as they are new, 



' The proof of this valuable theorem is extremely simple. It rests on 

 the following principles : — 



(i) That any number which is a common measure to two cyclotomes to the 

 same base must divide the Fermatian to that base whose index is their 

 greatest common measure. This theorem need only to be stated for the proof 

 to become apparent. 



(2) That any cyclotome is contained in the quotient of a Fermatian of 

 the same index by another Fermatian whose index is an aliquot part of the 

 former one. The truth of this will become apparent on considering the f ^rm 

 of the linear factors of a cyclotome. 



Suppose now that any prime number, k, is a common measure to two cyclc- 

 tonies whose indices are PQ, PR respectively, where Q is prime to R, and 



e^Q - I; 



whose common base is ©. Then ^must measure — I and alsa 0P _ j 

 it will therefore measure Q, and similarly it will measure R ; therefore k = i 



©PQ - I . 



— is unity, and 



[unless Q = r or R = i ; for suppose Q = i, then — 



no longer contains the core of 0"Q — I]. Hence/?; being contained in R can 

 only be an internal factor to one of the cyclotomes (viz. the one whose index 

 is the greater of the two). [See footnote at end.] 



The other theorem preceding this one in the text, and already given in the 

 " Excursus," may be proved as follows : — 



Let A, any non-unilinear function of P, the index of cy 1 to ne X, bs a 

 divisor thereto. Then, by Euler's law, there exists some number, yU, such 



P 

 that k divides Xij. — I, but the cyclotome is contained algebraically in 



P ; hence k must be continued in /^i, and therefore in P. Also, k will 



X^ - I 



I X^ -I I 



be a divisor of .^* - I and of p , which contain X'^ — I and X respect- 

 x^ -I 



X^ -I 

 ively ; consequently, if k is odd, ,4^ will not be a divisor of p , and 



X^ - I 

 a fortiori not of X- [A proof may easily be given applicable to the case 



0f^ = 2.] 



Again, let P = Q^', where Q does not contain k. Then, by Fermat's 

 theorem, xk''=x [mod. k]. and therefore k divides X*^ — I ; but it is prime to 

 Q. Hence, by what has been shewn, k must be an external divisor of this 

 function, and consequently a unilii ear functi,n of Q. Thus, it is seen 

 that a cyclotome can have only one internal divisor, for this divisor, as has 

 been shown, must be an element of the index, and a unilinear function of 

 the product of the highest powers of all the o.her elements which are 

 contained in the index. 



For an extension of this law to "cyclotomes of the second order and 

 conjugate species," see the " Excursus," where I find the words extrinsic 

 and intrinsic are used instead of external and internal. 



^ A reduced Fermatian obviously may be resolved into as many cyclotomes, 

 less one, as its index contains divisors (ur.ity and the number itself as usual 

 counting among the divisors). But, barring the internal d. visors, all these 

 cyclotomes to a given base have been proved to be prime to one another, 

 and, consequently, there must be at least as many distinct prime divisors as 

 there are cyclotomes, except in the very special case where the base and 

 index are such that one at least of the cyclotomes becomes equal to its 

 internal divisor or to unity. It may easily be shown that this case only 

 happens when the base is - 2 and the index any even number, or when the 

 base is + 2 and the index divisible by 6 ; and that in either of these cases 

 there is f nly a single unit lost in the inferior limit to the number of the 

 elements in the reduced Fermatian. 



has taken its origin in the felt necessity of proving a purely 

 negative and seemingly barren theorem, viz. the non-exist- 

 ence of certain classes of those probably altogether ima- 

 ginary entities called odd perfect numbers : the moral 

 is obvious, that every genuine effort to arrive at a secure 

 basis even of a negative proposition, whether the object 

 of the pursuit is attained or not, and however unimportant 

 such truth, if it were established, may appear in itself, is 

 not to be regarded as a mere gymnastic effort of the 

 intellect, but is almost certain to bring about the 

 discovery of solid and positive knowledge that might 

 otherwise have remained hidden.^ J. J. Sylvester. 

 Torquay, February 11. 



LORD RAYLEIGH ON THE RELATIVE 

 DENSITIES OF HYDROGEN AND OXYGEN:^ 



THE appearance of Prof. Cooke's important memoir 

 upon the atomic weights of hydrogen and oxygen,'* 

 induces me to communicate to the Royal Society a notice 

 of the. results that I have obtained with respect to the 

 relative densities of these gases. My motive for under- 

 taking this investigation, planned in 1882,^ was the same 

 as that which animated Prof. Cooke— namely, the desire 

 to examine whether the relative atomic weights of the two 

 bodies really deviated from the simple ratio i : 16, 

 demanded by Prout's law. For this purpos a knowledge 

 of the densities is not of itself sufficient ; but it appeared 

 to me that the other factor involved, viz. the relative 

 atomic volumes of the two gases, could be measured with 

 great accuracy by eudiometric methods, and I was aware 

 that Mr. Scott had in view a redetermination of this 

 number, since in great part carried out.'' If both in- 

 vestigations are conducted with gases under the normal 

 atmospheric conditions as to temperature and pressure, 

 any small departures from the laws of Boyle and Charles 

 will be practically without influence upon the final number 

 representing the ratio of atomic weights. 



In weighing the gas the procedure of Regnault was 

 adopted, the working globe being compensated by a 

 similar closed globe of the same external volume, made 

 of the same kind of glass, and of nearly the same weight. 

 In this way the weighings are rendered independent of the 

 atmospheric conditions, and only small weights are re- 

 quired. The weight of the globe used in the experiments 

 here to be described was about 200 grammes, and the 

 contents were about 1800 c.c. 



The balance is by Oertling, and readings with successive 

 releasements of the beam and pans, but without removal 

 of the globes, usually agreed to one-tenth of a milligramme. 

 Each recorded weighing is the mean of the results of 

 several releasements. 



The balance was situated in a cellar, where temperature 

 was very constant, but at certain times the air currents,, 

 described by Prof. Cooke, were very plainly noticeable. 

 The beam left swinging over night would be found still in 

 motion when the weighings were commenced on the 

 following morning. At other times these currents were 

 absent, and the beam would settle down to almost absolute 

 rest. This difference of behaviour was found to depend 

 upon the distribution of temperature at various levels in 

 the rooms. A delicate thermopile with reflecting cones, 

 was arranged so that one cone pointed towards the ceiling 



' Since receiving the revise, I have no' iced that it is easj' to prove that 

 the algebraical resultant of two cyclotomes to. the same base is unity, ex- 

 cept when their indices are respectively of the forms Q(/tQ + i)''' and 

 Qt-^Q + i)S where (/tQ + i) is a prime number, .ltd Q any number (unity 

 not excluded), in which case the lesultant is kf^ -|- i. This theorem sup- 

 plies the raison raisonnee of the proposition proved otherwise in the first 

 part of the long footnote. 



^ A Paper read at the Royal. Society on February 9. 



3 "The Relative Values ot the Atomic Weights of Hydrogen and 

 Oxygen," by J. P. Cooke and 'i". W. Richards, Amer. Acad. Proc, vol. 

 xxiii., 1887. 



4 Address to Section A, British Association Report, 78?2. 



5 " On '.he Composition of Water by Volume," by A. Scott, Roy. Soc. 

 Proc., June 16, 1887 (vol. xlii. p. 396). 



