TRANSACTIONS OF SECTION A. 267 



into the complex formed by the centre of the triangle and an equiharinonic cubic, 

 whose cusps are at infinity, 



(-p + q + r) (p-q + r) {p + q-r)=±pqr. 

 10. Diagrams were exhibited of the bounding curves, or loci of the satellites of 

 the foci in°groups of confocal cubics, when the foci stand at the angles of equi- 

 lateral, isosceles, and scalene triangles, both acute-angled and obtuse-angled. Com- 

 plete sets of critical bitangential and inflexional cubics, with their companion 

 curves, were also exhibited, to illustrate every possible variety of class-cubic in 

 each family of groups. 



8. On a Modification of the Law of Facility. 

 By Donald M'Alister, B.A., B.Sc. ., 



Suppose we prepare a series of tints of grey, composed of varying proportions of 

 black and white, and arrange them in regular gradation of depth so that to the 

 eye the successive terms of the series differ by equal amounts. Then experiment 

 and observation, summed up in the Law of Fechner, show us that the series _ of 

 numbers which express the percentages of black (or of white) in the successive 

 tints form a geometrical series. If now a person tried to match a grey tint which 

 he had seen, he would be liable to error. By the ordinary principle, in any large 

 number of such fallible matches, we deem equal departures from the truth to be 

 equally probable, and take the mean of all the estimates as the best value of the 

 true tint which we can derive from them. But the previous considerations show us 

 that the ' mean ' must be not the arithmetic mean, but the geometric mean. For ex- 

 ample, tints containing 4, 8, 16, parts of black will seem equally graded. It is as 

 likely, therefore, that, the true tint being 8, an estimate ;16 shall be made as an 

 estimate 4. We should make a mistake if, having only these two estimates before 

 us, we inferred that the AM. or 10 was most probably the truth, and not the 

 GM. or 8. 



This particular example is the type of a large number of cases connected with 

 fallible estimates, and of many statistical series where there is reason to believe that 

 a ' geometric mean ' gives a truer average or representative than the ordinary arith- 

 metic mean. It becomes of importance to inquire what modification must be made 

 in the Law of Facility. This law purports to represent the distribution of aberrant 

 measures round the mean. And it is well known that the assumption that the 

 AM. is the most probable value leads to the expression of the Theory of Errors, 

 viz., y = e -fc 2 ^ 3 , x being the measure and a the mean. What law follows from 

 the assumption that the GM. is the most probable mean ? This is the gist of the 

 reasoning and the problem which Mr. Francis Galton laid before me some time 

 since. I propose here merely to state my answer, leaving the proofs and the 

 development of the theory to another occasion. 



If x (as before) be the measure, a the geometric mean, the (infinitesimal) pro- 

 bability that x is the estimate made is proportional to 



exp(-^(log£) 2 ), 



where ' exp ' is brief for ' e to the power of? 



If the question be modified, as suggested by the ordinary theory, and it be asked 

 'What is the probability of an estimate lying between the close limits x and 

 x + dx ? ' The answer is — 



h I is/I 3\ 2 \&c 



_exp(-A-(log Ca ))-. 



In both cases h is a constant depending on the general closeness of the measures 

 which, as in the ordinary theory, we may call the 'measure of precision' or 

 ' weight? 



The matter has statistical and physiological bearings of great interest, and I 

 believe of some practical value. 



