From these experiments it follows that all insulators offering a resistance up to about 1 millim. S. unit can be detected by the fingers, and those above 1 millim. and under 8 millims. can be unmistakably detected by the tongue. It appeared also that tongues of different persons were equally sensitive, since several persons (Europeans and natives) acknowledged the known acid taste, even through the insulator No. 9, having 8.2 millims. S. units resistance.

The highest limit of the method could, of course, be increased by filling the revolving bobbin of the magneto-electric machine with much finer wire and increasing the number of permanent magnets; however this will be scarcely necessary, because it seems to be a fact that if an insulator has more than about 8 millims., the resistance is generally so high as to be practically infinite, and therefore a greater sensitiveness of the instrument would only complicate the method.

As it is intended that the tester himself should turn the handle of the magneto-electric machine, he has it entirely in his power to regulate the strength of the induction-currents by turning faster or slower; and as, besides this, he always begins the testing by at first sending the currents through his fingers, no severe shocks can occur to him in the subsequent operation.

The method has also a safeguard in itself against carelessly rejecting good insulators, because the tester will certainly be careful in having the insulator properly cleaned before testing it, in order to avoid severe shocks.

There can also be scarcely any doubt that the tongue is the

best detector in this particular case, because it is sufficiently sensitive, never gets out of order, and indicates almost momentary currents; it is besides the cheapest instrument that could be used.

Note. This method may also with advantage be used for detecting bad joints in a telegraph line. It is then only necessary to connect the two ends of the joint to the two terminals of the magneto-electric machine in such a way that the body of the tester acts as a shunt to the joint.

A joint which offers a resistance of not less than 5 S. units allows a current to pass sufficiently strong to be detected by the tongue; but if the joint has a resistance of more than 200 S. units, the current passing is strong enough to be felt already by the fingers of the tester.

XIV. Investigation of the Law of the Progress of Accuracy, in the usual process for forming a Plane Surface. By GEORGE BIDDELL AIRY, Astronomer Royal*.

IN

N order to form a plane surface, it is usual to take three surfaces (which I shall call A, B, C), and to grind A with B till they fit together, then to grind B with C, then to grind C with A. (I shall call each of these grindings a rub, and the system of three rubs an operation.) And the problem which I propose is, to find the deviation of each surface from a plane, after n operations, expressed as a function of n.

I shall assume that at each rub the surfaces are worked into perfect contact. Putting A and B for the prominences of special parts of the surfaces of A and B above a mean plane, and putting A' and B' for the state to which they are changed after the rub, and remarking that convexity of one corresponds to concavity of the other, A'+B' must = 0. I shall also assume that equal portions are worked off the two surfaces-that is, that A-AB-B'. These equations give

A'A-B, B'=−A+†B.

Considering now the effect produced by the first of all the operations, the expressions for the prominences are as follows:

completing the + A-B-C-A+4B-C −}A+}B+‡C

first operation.

* Communicated by the Author.

I shall not delay longer on this first operation, except to make the following remark. At the end of this first operation, the new values of A and C are equal but with opposite signs; and it is evident that the same remark will apply to the result of every succeeding operation. But it did not apply to the values A, C with which we started. We must therefore consider these values obtained after the first operation as the beginning of the symbolical series; and we may call them Ao, Bo, - Ao

Suppose now that we have gone through n operations, forming the values A, B,, -An; and that we examine the effect of the n+1th operation. We have

After first rub+A-BA, +B

-A

-A

After second rub. +A-B+4A, +4B2-A, -4B,

Multiply the second by an indeterminate constant p, and add it to the first; then

(An+1+p. Bn+1)= ( &; + ¦
¦ p)^n + ( − ¦
¦ + ¦
¦ p)Bn,

8

8

ing value of +p. Then the equation is

8

(An+1+p. Bn+1)=9 . (An+p. Bn),

of which the solution is

An+p. B2 =E.qn+ß;

n

where E is a constant, to be determined so as to satisfy initial circumstances, and where ẞ may be fractional.

As the equation for p will be a quadratic, there will be two

values of p, two corresponding values of q, and two permissible values of E and of B; and the solutions will be

An+p". Bn=E". (q′′)~+ß" ;

from which A, and B, may be found.

Forming the equation for p, we find 2p2+p+1=0; the roots of which may be expressed in the form

p'=−√/{/ · {√}{} −√=1.√}},

It

appears

here that (p')=q", and (p")3=q' ; and this is verified in the following manner. From the equation for p just employed,

Therefore (p)=-p'. But, in the second term of the

equation,

p'+p"=-1, p'=-1-p",

8

and therefore (p)=q". Similarly (p")3=q'.

The expressions for p and q may be more conveniently put in the following form:

and

q=√. {cos 3a +3π+ √1. sin 3x+3π},

(9')"+P' = ( '}')'"'+'"'. {cos (n + B' · 3a +3π)

+1.sin (n+B' · 3a +3π)},

2. {cos (n+B". 3a +3π)

+1. sin (n+B". 3a +3π)}.

In algebraic generality there is nothing to prevent E from consisting of two terms, one being imaginary. But such an expression could be put under the form of a cosine and a sine with imaginary factor, and its effect would be simply to add a constant to the constant B, and nothing would really be gained in generality. And, upon attempting to solve the equations for A, and B, it would be found immediately that the condition of real values for A, and B, requires that E' and E" be equal, and that ' and 8" be equal. The equations are therefore to be used in this form :—

An+p'. B2 =E.(q′)”+ß,

An+p". B2 = E. (q′′)n+ß.

n

=-E. ( 1 ) + 2√ = 1. sin (n+ß • 3a +3π)

√1.sin (a +π)

sin (+) (+)+ sin (7+8. 3a +3π).

=

Second, multiplying the first equation by p" (q) and the