138 



NATURE 



[April i, 1920 



disappeared, leaving one or two longish ones. The 

 colour was mostly white, but sometimes reddish in 

 parts, especially nearly due north. 



"A curious feature was an oblique band of light, 

 which came and went across near the summits of the 

 vertical beams. I do not think this was a belt of 

 illuminated cirrus, as its brightness seemed to vary 

 independently of the vertical beams, but it is possible 

 it may have been. The lights had diminished con- 

 siderably by about 3.45 a.m., but had brightened 

 again, though slightly, when I looked out a few 

 minutes later. I do not know what time the display 

 ended." A. L. Cortie. 



Stonyhurst College Observatory, March 29. 



Methods of Approximate Integration and of 

 Computing Areas. 



The formulae which Mr. Percival gives in Nature 

 for March 18 for approximate integration are well 

 known, but there are one or two points in connection 

 with them which are frequently overlooked, especially 

 by writers of books on mathematics for engineers. 



(i) The areas bounded by curves the equations of 

 which are of the form 



y = a-\-hx + cx'->r . . . +kxn 

 can be obtained from the values of 2m +1 equi- 

 distant ordinates, not only when n = 2in, but also 

 when n = 2m+i. That this is so is seen most easily 

 by taking the origin at the centre of the range of 

 integration and noting that 



J —h 



For example, Simpson's first, or three-ordinate, 

 rule gives the area of the cubic 



with perfect accuracy, and for this purpose his second, 

 or four-ordinate, rule is in no way superior. 



(2) By a very small change in one of the coefficients 

 Weddle threw the seven-ordinate formula (No. 6 in 

 Mr. Percival 's letter) into the very convenient form 



A = ^[ti + J3 +/6 +^r + 5 (J2 + Je) + 674] • 



The loss of accuracy which the change involves is 

 exceedingly small. 



(3) Formulae based upon the assumption that the 

 boundary curve can be represented by an equation of 

 the form above stated give unsatisfactory results 

 when the actual boundary has tangents at right angles 

 to the A--axis. This is really the reason why none of 

 the results obtained by Mr. Percival in applying his 

 formulae to the quadrature of a circle possess a 

 higher degree of accuracy than that represented by 

 the admission of errors of the order of i per cent. 



If we suppose the curve to cut the axis of x at 

 right angles at the origin, it is better to assume that 

 it can be represented by y:=ax\ + hx in the neigh- 

 bourhood of that point. 



If y,, y^ be the ordinates at x = h, x = 2h, the area 

 bounded by the curve, the axis of x and the ordinate 

 y2 is given by 



The much higher degree of accuracy resulting from 

 the employment of this formula may be illustrated by 

 applying it to Mr. Percival 's example of the quadrant 

 of a circle. 

 The seven ordinates are : — 



yo = o 3/4 = 0-9428090 



yi =0-5527708 y, = 0-9860133 



y2= 0-7453560 ye = i 



y, = 0-8660254 



NO. 2631, VOL. 105] 



Using the above formula to find the area between 

 the ordinates y^ and yi, and Simpson's first formula 

 for the part between yj and y,, we obtain the value 

 0-7853871. The true value is 07853982; hence the 

 percentage error is only 00014, which compares very 

 favourably with the errors ranging from 0-8 to 1-34 

 per cent, obtained by using the usual formulae for 

 the whole range. 



Mr. Percival's example clearly shows that when the 

 curve has a tangent at right angles to the axis, no 

 material reduction in error is attained by using 

 formulae with a larger number of ordinates. The use 

 of Simpson's formula over ordinary ranges and of the 

 formula given above in the neighbourhood of such 

 tangents will prove much less laborious and far more 

 accurate. J. B. Dale. 



King's College, Strand, March 22. 



In Nature of March 18 Mr. A. S. Percival gives 

 an example (the quadrant of a circle) in which Simp- 

 son's rule (sometimes called his first rule) is more 

 accurate than the "three-eighths" rule, and he 

 remarks: "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." Permit me to point out that this infer- 

 ence is not valid, and is based on the almost universal 

 illusion that Simpson's rule is correct to the second 

 order only, i.e. for the parabola 



y — a+bx + cx'^. 



It is easy to show by simple integration that Simp- 

 son's rule holds to the third order, i.e. for all cubics 

 of the form 



y = a+hx+cx' + dx^, 



passing through the three chosen points. It is thus 

 precisely accurate, not only for the parabola, but also 

 for a singly infinite number of curves passing through 

 the three points, even if an inflexion occurs. 



One would therefore expect (which I believe to be 

 the case) that where both rules can be applied (e.g. if 

 there are seven ordinates) Simpson's rule would be 

 more accurate than the "three-eighths" rule, which 

 is precisely true only for a single curve passing 

 through four consecutive points. 



In some cases, when the gradient is not rapid, 

 Simpson's rule is highly accurate. Dr. Lamb 

 ("Infinitesimal Calculus," p. 278) gives an example 

 in the cvalution of n- to six decimal places from the 

 equation 



f^ a :t _n 



by taking ten equidistant values for x, but he does 

 not notice the illusion to which I refer. I am sur- 

 prised that such a simple and easily tested truth 

 should so long have escaped the notice of many expert 

 mathematicians. R. A. P. Rogers. 



Trinity College, Dublin, March 20. 



Gravitational Deflection of High-speed Particles. 



The result mentioned by Mr. Leigh Page and 

 verified by Prof. Eddington (Nature, March 11, p. 37), 

 that the gravitational effect on a particle travelling 

 radially is a repulsion if the speed exceeds 1/^/3 times 

 the light-velocity, is given by Hilbert in the Gottinger 

 Nachrichten for 1917. The same paper contains in- 

 teresting remarks on the path of a particle or light- 

 pulse moving spirally round the gravitation centre. 



Hvmers College, Hull. H. G. Forder. 



