March i8, 1920] 



NATURE 



71 



I 



When m = o, the ordinates with m as a part of their 

 subscript are omitted in all but the first term. 



Example. — Suppose the base line be divided 

 six^ equal intervals (H = i), and the value of 

 ordinates be 



into 

 the 



y,=o 



y» =07453560 



y, =09428090 



^2 = 0-5527708 

 y^ = 08660254 

 >-, = 0-9860133 



As seven ordinates are given, v^e may use any one 

 of the rules (6), (3) which is adapted to 4 + 3m 

 ordinates, and (2) which can be used when 3+ 2m 

 ordinates are employed. We should expect to get a 

 more accurate result when the higher-order formula 

 is employed, and this we shall find to be correct. 

 The values given refer to the quadrant of a circle, 

 so that the true value is n-/4, or 0-78539816 . . . 



By method (6), putting- w = o, the result is 0-7791866, 

 i.e. 07972 per cent, too small. 



By method (3) the result is 0-7758061, or 1-342 per 

 cent, too small. 



By method (2) the result is 0-777531, or 1-063 per 

 cent, too small. 



This result is curious, and shows that a small arc 

 of a circle approaches more nearly to a small arc of 

 a parabola than to a small arc of any cubic curve, 

 but it will be noted that method (6) gives a much 

 more accurate result. 



We may, however, use a combination of the above 

 rules; for instance, we may take five ordinates by 

 rule (4) and the remaining two intervals by rule (2). 

 As the first three ordinates increase more rapidly than 

 the last three, we should naturally leave the last 

 three to be dealt with by rule (2). In this way a 

 result of 07784954, or a defect of 0-0069027, or an 

 error of only 0-88667 per cent, is obtained. Had we 

 reversed the order and used Simpson's rule for the 

 first two intervals, the defect would have been 

 0-0078447, or an error of 1-0102 per cent. 



In conclusion, it may be stated that if the nature 

 of the curve is unknown a more accurate result will 

 always be obtained by using the highest-order formula 

 that can be used with the given number of ordinates. 

 If two different formulae are used, it has just been 

 shown that the most accurate result is obtained when 

 the higher-order formula is used for that part of the 

 curve in which the variation of the ordinates is the 

 greater. If the curve be a parabola, an absolutely 

 accurate result is obtained by using only three 

 ordinates by means of method (2). 



It may be thought that plotting the curve and 

 estimatingr its area mechanically by means of a plani- 

 meter will be always the best and speediest method to 

 adopt, but this is by no means the case. It often 

 takes far less time to calculate, say, thirteen ordinates 

 and to use method (6) than to trace the curve. 



A. S. Percival. 



Westward, Newcastle-upon-Tyne. 



An Electronic Theory of Isomerism. 



The interesting suggestions made bv Mr. W. E. 

 Garner in Nature for February 19 with regard to a 

 possible explanation of the isomerism of certain 

 organic compounds may be examined from a different, 

 but perhaps simpler, point of view by employing the 

 "ring electron " or "magneton " of Mr. A. L. Parson. 

 The electron is looked upon as a circular anchor ring 

 of negative electricity rotating about its axis at a 

 high speed, and therefore behaving like a small 

 magnet. In connection with atomic and molecular 

 NO. 2629, VOL. 105] 



numbers 1 have directed attention elsewhere to the 

 "rule of ei^ht," according to which a difference of 

 8 or a multiple of 8 is frequently found between the 

 numbers of the unit electric charges associated with 

 analogous atoms or molecules. In the theory of the 

 "cubical atom" put forward by Prof, tiilbert N. Lewis 

 and developed by Dr. Irving Langmuir, one of the 

 most stable configurations for the atomic shell is 

 that in which eight electrons are held at the corners 

 of a cube. The single bond commonly used in 

 graphical formulae involves two electrons held in 

 common by two atoms (Fig. i); the double bond 

 implies that four electrons are held conjointly bv two 

 atoms (Fig. 2) Or if the pair of electrons be regarded 

 as the most stable grouping of all, it may be, as 

 Lewis and Langmuir suggest, that the pairs of elec- 

 trons held in common bv two atoms are drawn closer 



Fig. 3- 



together by the magnetic attraction between them. 

 Dextro- and laevo-rotatory forms of a compound might 

 then be represented as mirror images as in Fig. 3. 

 The letters N and S in this diagram may be taken 

 to represent the polarity of the external face of the 

 ring electron. 



Mr. Garner suggests the possibility of the existence 

 of a large number of optical isomerides amongst 

 organic compounds, but the view here put forward 

 does not lead to that conclusion ; on the contrarv, it 

 seems to give exactly the same number of isomerides 

 as the ordinary structural formulae. It is true that 

 it is possible to reverse in the diagram the magnetic 

 polarity of one x>r more pairs of electrons, but even 

 if the arrangements so obtained were stable, it is 

 doubtful whether they would represent different iso- 

 merides. It would not be possible to explain the 

 phenomenon of free mobility about a single bond 



