April 1, 1898.] 



KNOWLEDGE 



76 



In how many different ways may the 

 square be read from 1 to 11 consecutively ? 



figures in the 

 Answer, 252. 



The smallest square that can be arranged is one of tim 

 1 2 



dimensions, viz., 



2 3 



and it is at once obvious that the 



figures in it can only be read two different ways. 



It is equally obvious that in a square of three dimen- 

 1 "2 3 

 sions, viz., 2 3 4, that the figures can only be read in six 



3 4 5 

 different ways. 



But in a square of four dimensions they may be read 20 

 different ways ; in one of five dimensions, 70 ways ; in one 

 of six, 252 ways ; in one of seven, 924 ; and so on, the 

 numbers forming a series in geometrical progression of 

 which 2 is the first term, and 3 the common ratio, Init after 

 till' SECOND ti-nii raih swrcxsiir tinii is phi-i a fnictinn of the 

 preci'din;/ term. 



It IS this constantly recurring fraction, and the form it 

 takes, which, to me, are the novel features of the series. 



Where n is any whole number greater than 2 the nth 

 term of the series always takes the form — 



nth term = 3 («-l term) -I- " '-' (« — 1 term). 

 (The nth term answers to a square of n + 1 dimensions). 



The series therefore runs as follows, viz. : — 

 1st term ... ... ... ... = 



= (3x2) 



= (3x6) + 1(6) ... 

 = (3x20) + 1 (20).-- 

 = (3x70) + 1(70) 

 = (3 X 252) + A (252) 



2 



6 



20 



70 



252 



924 



2nd 

 3rd 



4th 

 5th 

 6th 

 and so on, 



It will be seen that while the fraction in each term of 

 the series takes the form of " ~ " that when the mwirrual 

 value is given to each one, according to the term m which 

 it appears, we have a series of fractions whose numerators 

 and denominators each form a series of numbers in 

 (trithmHical progression, the numerators beginning at unity 

 and the denominators at 3, the common difference in each 

 case bemg 1, viz., i, f, f, f, 4, |, |, . . . and so an., ad 

 iiijinitioii, the successive fractions constantly increasing 

 but never reaching the value of -J. 



The numbers carried to the 10th term of the series are 

 2, 6, 20, 70, 252, 924, 3432, 12,,S70, 48,620, 184,756, and 

 so on. 



Has such a series of numbers any mathematical designa- 

 tion ■? Faithfully yours, W. Staniforth. 



Upperthorpe, Sheffield, March 13th, 1893. 



[I am not aware of any distinguishing name for 

 Mr. Staniforth's series. — A. C. E.] 



THE CONSTITUTION OF GASES. 



By J. J. Stewart, of Emmanuel Collecie, Canibridi/e. 

 {Continued from page 56.) 



TO find the pressure on the sides of a vessel due to 

 the contained gas, imagine the containing vessel 

 to have the shape of a cube, and let / be the 

 length of an edge, V the speed of each molecule 

 of gas, and n their number. Now a cube has 

 three pairs of faces, each pair being at right angles to 



the other two pairs ; and the velocity of a particle may 

 be considered as made up of three velocities at right 

 angles to each other (call these component velocities 

 u, V, ir, and V the actual velocity, then the relation 

 between the velocity V and its components is given by the 

 equation V- = «- + c- + »•'). Thus we may suppose one- 

 third of all the gas particles to be moving perpendicularly 

 to each pair of faces of the cube, and the number of im- 

 pacts made per second by any one particle on any one face is 

 .7| , for the particle moves over twice the length of the cube 

 between each of its impacts on the same face. As there are 

 n particles present the total number of impacts on one face 

 is 'g^- Now, as at each impact the speed of each particle 

 is reversed — after an impact the molecule is moving with 

 speed unchanged in magnitude but reversed in direction — 

 a-nd as the mass of each molecule is - { the whole mass of 

 gas divided by the number of molecules), the measure of 

 the impact is " '^-^- To get the pressure from these data 

 we must multiply the value of the impacts by their 

 number per second, and to express the pressure in terms 

 of unit area we must also divide by the area of the face, 

 i.i\,l-. Thus the pressure IS „ x ^ x ,, := .y^- fjut 

 the quantity f; is the mass of gas divided by its volume, 

 i.e., its density, so that the pressure is ^rd of the density 

 of the gas multiplied by the square of the velocity of its 

 constituent molelcules ; or calling the density of the gas D, 

 pressure = ^ D \'~ — p. From this expression .Joule 

 obtained the velocity of the molecules of the gas, for the 



mean square of the velocity V- = -^ where /> = the 

 pressure of the gas, and D = its density. For example, 

 take the case of hydrogen, whose density expressed in 

 pounds per cubic foot is -0050, at ordinary atmospheric 

 pressure, which is about 2115 pounds weight on the square 

 foot. Let the gas, moreover, be at the fi-eezing point. 

 The pressure in pounds weight per unit of area must be 

 multiplied by the intensity of gravity in feet per second per 

 second {i.e., by 32-2) in order to bring the measurements to 

 the absolute system. Then we have for the square of the 

 velocity V- = ""* '^^^' "" ' = 36,500,000 very nearly, and 

 the square root of this = V = 6040 feet per second. 

 That is, the molecules of hydrogen gas at the freezing 

 temperature are moving about with a velocity of over a 

 mile per second. This speed is far in excess of that of a 

 bullet when on its path from a gun. 



It must not be supposed that most of the molecules have 

 this particular velocity, nor that a single molecule retains 

 this speed for a lengthened period. At ordinary pressures 

 the hydrogen particle which may happen to possess the 

 above average speed after one of its encounters with a 

 neighbour has scarcely well started in its new path before 

 both its direction and velocity are changed by a fresh 

 collision. But the above quantity, 6040 feet per second, 

 is an important and characteristic property of hydrogen, 

 and comes into account whenever we have to consider the 

 average behaviour of masses of the gas as effected by the 

 molecular velocities ; for instance, in considering the 

 rate of diffusion of hydrogen through a porous 

 diaphragm or a crack in a glass vessel. Other phenomena 

 (especially chemical onesi cannot be explained without 

 supposing some of the particles of hydrogen to have a much 

 higher velocity than this average one. As another 

 example consider the case of oxygen, and let us now 

 calculate the values, using the centimetre as the unit of 

 length, the gramme as the unit of mass, and the second as 

 the unit of time — these units being now almost universally 



