400 “TRANSACTIONS OF SECTION A. 
to secure the most favourable result in a limited time, a criterion for ‘best 
magnitudes,’ previously proposed,’ is here further considered, and illustrated 
by an application to the interferometer. 
In regard to computation, the availability of an automatic device for 
linear least-square adjustment * makes it now desirable to have some means of 
throwing an assumed relation into linear form without disturbing the relative 
weights of the observations. A general formula for doing this is here proposed, 
and applied to the determination of thermal expansion coefficients. 
Finally, the investigation of economy of time in taking the observations 
themselves leads to two distinct problems: first, that of the division of time 
amongst the components of an indirect measurement; second, that of the best 
grouping of observations in determining any one quantity. 
The solutioa of the first problem comes out in terms of three data—namely, 
the relative precision of, and the‘relative time consumed in, a single observa- 
tion on the respective components; together with the derivatives expressing 
the sensitiveness of the result with respect to the several components. Of 
these data the first two are postulated, while the third is implicitly contained 
in the equation which defines the measurement in question. The solution is 
independent of the existence of constant errors. 
The second problem consists in establishing the most profitable compromise 
between the extremes of (1) repeating a large number of readings under the 
same conditions (or on the same sample), in order to diminish the effect of 
observational errors; or (2) resting content with a lower precision on each 
determination, in order to cut down systematic errors by making numerous 
independent determinations (or by trying many different samples). The most 
economical nuraber of observations to make in any one group before stopping 
to change conditions (or to set up a new sample) in preparation for a new 
group is directly expressible in terms of two postulated data. There are, first, 
the ratio of the average observational error to the average systematic error 
anticipated; and, second, the ratio of the time required in preparing for a 
new group to the time used in a single observation. This result is independent 
of the total time available. 
The first problem is illustrated by the division of time in a gravity deter- 
mination by Kater’s pendulum; the second, by the determination of the heat of 
combustion of coal from a series of samples. 
A combination of the two problems may also arise. The solution is equally 
straightforward. 
Throughout, the object of the paper has been to establish certain general 
principles governing the accuracy attainable in physical measurements, inde- 
pendently of the particular apparatus or process in question. 
8. Some Remarks on Waring’s Problem. 
By Professor J. EK. A. SrEGGauu. 
9. On a System of Spherical or Hyperspherical Co-ordinates. 
By T. C. Lewis. 
If any hypersphere be taken, let the co-ordinate of a point with reference to it be 
», where p is the radius and § the square of the tangent from the point considered. 
In n-space take any system of »-++-2 fundamental spheres such that each one cuts all 
the others orthogonally. Their centres are at the vertices and orthocentre of an 
orthocentric figure. 
Let the co-ordinates of a point be 2, @,..., 2ryo 
They are connected by the two relations :— 
Miia we et Se 
S710 ie Bard ES Dy (ii) 
1 Jour. Wash. Acad. Sci., vol. i., 1911, p. 187. 
* Tbid., vol. ili., 1913, p. 296. 
