In this case, if we put c = 1, ẞ and μ are the Stokes functions corresponding to a and A. If the level surfaces of the harmonic space functions, V and W, are surfaces of revolution about two different straight lines in the xy plane, the functions a and A which represent the values of V and W in this plane do not in general satisfy (31).

Graphical superposition of the lines of force in the xy plane due to an infinitely long, homogeneous cylinder of revolution parallel to the axis, and to a homogeneous sphere with centre in the plane, will not in general yield the lines of force in the xy plane due to a combination of the two masses.

If a and are harmonic, any linear function (but no other than a linear function) of a is harmonic, and any two linear functions of a and satisfy (31). There generally exist, however, non-linear functions of a and A which, although they are not harmonic, satisfy the condition. The functions (2-y2), (x2 + y2)", the second of which is not harmonic, obey (31), as do the harmonic pair (a2 — y2), log (x2 + y2).

As a simple example of the fact that a harmonic function and a function which is not even isothermal may satisfy the condition (31), we may consider (2 y2 - x2) and (y2 — x2).

The non-isothermal functions a2 - a y2, y2 — a x2, which are solutions of the equation

If a and A are any two solutions of the equation

where f(r) is any given function of x, the condition (31) is satisfied

and = f(x).

In general a and A must both be solutions of an equation of the

where P and Q are any functions of x and y such that dP/dy=dQ/dx.

The question whether if (a, ẞ) and (A, μ) are orthogonal pairs and (a+B+) is not an orthogonal pair, it is possible to find a function (B) of B, and a function (M) of μ such that (a+, B+M) shall form an orthogonal pair, has already been answered; for a and X must satisfy (31) in any case, and if they do this, may be determined from (29) and (30) and B and M from (17) and (20).

THE JEFFERSON LABORATORY,

HARVARD COLLEGE, CAMBRIDGE, MASS.

Proceedings of the American Academy of Arts and Sciences.

VOL. XLII. No. 8.- JULY, 1906.

CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL LABORATORY, HARVARD COLLEGE.

ON THE CORRECTION FOR THE EFFECT OF THE COUNTER ELECTROMOTIVE FORCE INDUCED IN A MOVING COIL GALVANOMETER WHEN THE INSTRUMENT IS USED BALLISTICALLY.

BY B. OSGOOD PEIRCE.