June i6, 1893.] 



SCIENCE. 



333 



thoroughly studied it will be found to belong to one of these re 

 cently differentiated genera. F. H. Knowlton. 



U. S. National Museum, Washington, D. 0. 



Mean Values. 



Miss Porter's kindly criticism (Science, June 2) of one point 

 in the article, "Sun-Heat and Orbital Eccentricity" {Science, 

 Apr. 28), gives occasion to say a work in regard to mean values. 

 Since the mean value of n quantities is the arithmetic mean of 

 their sum, it would appear at first glance as if the term were a 

 perfectly definite one; but if the quantities to be averaged are 

 successive values of a function of some variable, then clearly their 

 magnitudes depend not only on the nature of the function, but 

 also on the law of variation of the fundamental. Thus, suppose 

 we have the isotherm, p u = e, and wish to know the average 

 pressure between the volumes v = V-^ and v = v«. It is necessary 

 to make some assumption in regard to the variation of v. If its 

 increments are supposed equal, we understand by the " mean 

 Talue" of the pressure the average of the pressures corresponding 

 to the values of y. If the volume is assumed to depend in turn on 

 some other variable in such a manner that the abscissa-increments 

 are not equal, the mean value will now be the average of the new 

 series of pressure-ordinates corresponding to the series of values 

 of V arising under the second assumption. Evidently these two 

 means will in general be unequal, but one is just as properly the 

 " real average " as the other. The formula for mean value may 

 be derived by a method even simpler than the usual analytical one 

 as given by Williamson and Todhunter. Let it be required to 

 find the mean value of y where y = f {x) and x is an equicrescent 

 variable, li y =f {x) be treated as a curve referred to rectangular 



b 

 axes, / f {x) dx is the expression for the area. A, bounded by 



a 

 the X-axis, two ordiuates, and the portion of the curve intercepted 

 between the bounding ordinates. Let A = A', where A' is a. 

 rectangle whose base equals the base of A. Then the altitude of 

 A' is the average of the ordinates in A. For let 



2/i + 2/2 + yn _ .. 



: £/0! 



n 



the average of the series of ordinates. 



Then y^ + 1/3 + = 2/o H- 2/ -H . . . . on to 7i terms. 



Multiplying by A a; and summing, 



2 {y^ + y^ + . . . .) A X = ^ {y„ + y„ + . . . .) A x; 



or, making n indefinitely large, 



b b 



J y dx = ya J dx = y„(b — a), 

 a a 



b 

 / y dx = A, hence 2/0 (b — o) = A', 



But 



as having developed into this axis, whilst a circle of unit radius 

 with pole as centre has developed into a straight line parallel to 

 the axis, the radii-vectores keeping their normal position with 

 respect to the circle. In finding the mean value of the radius- 

 vector of an ellipse, d 6 being constant, the figure A has three rec- 

 tihnear sides: x = 0, x = v, and the X-axis. Its fourth side is 

 the curve, 



y= °(l-e') 

 1 + e cos X 



The base of the figure is n-; hence the mean value is 



and, since 6 — a is the base of the rectangle, A', y„ is it salti- 

 tude. 



For example, let it be required to find the mean pressure be- 

 tween the volumes ^i and v^. If the isotherm is pv^c, the 

 area, A, in this case becomes 



/" -dv = clog('^^; 



its base is y, — '^i > hence the mean pressure is 



V^ — Vi \v^/ 



This conception of mean values may be readily employed when a 

 curve is expressed in polar coordinates. If r=f(S), let x be 

 written for S and y for ;•. The Cartesian equation thus arising 

 furnishes a curve which sustains peculiar relations to the original 

 polar curve. The radii-vectores are taken out of their fan-shaped 

 arrangement and placed equi-distant and parallel, with their ex- 

 tremities on a common line, the X-axis. The pole may be viewed 



1 /■ an 



TT^, 1 + 



(l-fi^) 



dx = a (^ 1 _ I 



It will be seen that the area-method serves only when the ordi- 

 nates are equally distributed throughout the area A. In the 

 dynamical problem of the earth's mean distance from the sun it 

 is not 8 (or x) which is the equicrescent variable, but t, the time. 

 A must therefore be taken equal to 



/'rdt, 



for which r^f(t) must be given; but, as is well known, the 

 equation expressing the relation between r and t is transcendental 

 and cannot be written in the form r=f(t). Recourse must 

 therefore be had to other devices for finding the mean distance 

 when the problem is rendered kinematical by taking Kepler's 

 second law into account. Ellen Hayes. 



Wellesley, Mass. 



Iron and Aluminium in Bone Black. 



Will you kindly, in your next issue, print the following cor- 

 rections to my article on "Iron and Aluminium in Bone Black," 

 which has just reached me. 



Page 300, first column. In twentieth line (from the bottom of 

 page), after the word " permanent," insert, and boil. In nine- 

 teenth line (from bottom of page) remove the first two words : 

 "and boil." 



In twelfth line (from bottom of page) insert a decimal point 

 between 5 and at end of this line, for the figure must read 5.0 

 and not 50 grammes. 



Page 301, first column. In twentieth line (from bottom of page) 

 transpose after "iron." Instead of "aluminium, or the phosphate " 

 then should stand : or the aluminium phoi^phate predominates. 



J. G. WiECHMANN. 



New York, June 7. 



Estimated Distance of Phantoms. 



In Science of May 19, p. 269, Mr. Bostwick mentions the fa- 

 miliar experiment of binocular combination of regular patterns, 

 such as a tessellated pavement or figured wall-paper, by means 

 of ocular convergence, and states that in his case, although the 

 figures of the phantom thus formed appear smaller, yet contrary 

 to the statements of all other writers they do not appear nearer 

 but farther oflE than the real object. This seems to me inexplica- 

 ble if the phantom is really distinct. 



As I have very unusual facility in making such binocular com- 

 binations, I will very briefly describe an experiment of this kind. 

 I stand now looking down on the tesselated oil-cloth covering the 

 floor of the library. By ocular convergence I slide the two 

 images of the floor over one another in such wise as to combine 

 contiguous figures. After perhaps a brief interval of indistinct- 

 ness, the pattern appears with perfect clearness at half the dis- 

 tance of the floor and the figures of the patjern of half the real 

 size. The sense of reality is just as perfect as in the case of a 

 real floor at that distance. It seems to me as if I could rap it 

 with my knuckle. Taking now this phantom as a real object, by 

 greater convergence the plane can be brought up higher and 

 higher, until by extreme convergence it is brought within three 

 inches of the root of the nose and seen there with the greatest 

 distinctness in exquisite miniature, the figures being only one- 

 quarter inch in diameter. By relaxing the convergence a little, 

 the phantom-plane may be dropped and caught on lower and 



