unit of torsion. If now we insert these numbers in the above equation and solve for the values of p, which correspond to the various values of the torsion given in the third column of the table, the results given in the last column are obtained. The form of the equation shows that there are two values of p for any particular value of the torsion. If we differentiate (IV) with respect to p to get the value of p, for which the torsion is a maximum, we get p2= 1/A, or p=23.1. With the millimeter as the unit this is the same as p=0.116 mm, The value of the torsion for this value of Ρ is 254°. The curve indicates a maximum where p = 0.120 mm., and its value is 252°. The further discussion of equation (IV) is reserved until more and more accurate data have been collected. With the object of rendering the instrument more easily handled, and thus making it susceptible of greater accuracy, it is proposed now to modify the form of the suspended system in order to reduce its moment of inertia. When this has been done the results from this instrument should be quite as reliable as those from the other, and with the experience which has been gained both in constructing and handling the apparatus it seems quite certain that the desired object will soon be accomplished. Grateful acknowledgment is made to Professor Trowbridge for placing at my disposal all the resources of the laboratory, including the services of a mechanician 22 and of a glass-blower. 23 Without the assistance rendered by them the apparatus could not have been constructed. To Professor Hall, who first called my attention to the problem, I am indebted for much advice and encouragement. JEFFERSON PHYSICAL LABORATORY, HARVARD UNIVERSITY. 22 Mr. Thompson of the Physical Laboratory. Proceedings of the American Academy of Arts and Sciences. VOL. XLII. No. 7. — JULY, 1906. CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL LABORATORY, HARVARD COLLEGE. ON THE CONDITIONS TO BE SATISFIED IF THE ORTHOGONAL. THEMSELVES BY B. OSGOOD PEIRCE. CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL LABORATORY, HARVARD COLLEGE. ON THE CONDITIONS TO BE SATISFIED IF THE SUMS OF THE CORRESPONDING MEMBERS OF TWO PAIRS OF ORTHOGONAL FUNCTIONS OF TWO VARIABLES ARE TO BE THEMSELVES ORTHOGONAL. BY B. OSGOOD PEIRCE. Presented March 14, 1906. Received April 12, 1906. IF 1 (x, y), 2 (x, y) are the potential functions due to two columnar distributions of matter the lines of which are perpendicular to the xy plane, and if y1 (x, y), 2 (x, y) are conjugate to 1 and 2, respectively, the families of curves obtained by equating 1 and 1⁄2 to parameters, are lines of force of the two distributions. Moreover, 1+2 is the potential function due to a combination of the two distributions, and the function 1+2 equated to a parameter gives the corresponding lines of force. The fact that if ($1, 41) are any pair of conjugate functions and (2, 2) any other such pair, the functions (a + b‡ a 41 + b ¥2) are also conjugate with similar facts for other classes of functions-lies at the foundation of the graphical methods so successfully used by Maxwell' and by others in drawing equipotential lines, and lines of force or flow, due to combinations of simple elements. If ($1, 41) are merely a pair of orthogonal functions and (2, 2) another such pair, it is generally not true that ($1 + $2, 41 + ¥2) are an orthogonal pair: thus (x, y), (x2 + y2, y/x) are pairs of orthogonal functions, but x + x2 + y2, y + y /x are not orthogonal. In certain classes of physical problems one encounters potential functions which are not themselves harmonic and the lines of which are not possible lines of any harmonic function, and it is often de 1 Maxwell, Treatise on Electricity and Magnetism, Vol. I, Ch. VII. Minchin, Uniplanar Kinematics, § 112. See also P. W. Bridgman, The electrostatic field surrounding two special columnar elements, These Proceedings, 41, 28. sirable in cases where the analytical processes become too complex, to determine graphically the forms of lines of force or flow due to a combination of two simple elements. This note discusses briefly the conditions under which the ordinary method of procedure is possible. Let (a, ẞ) and (A, μ) be two pairs of orthogonal functions of the two variables (x, y), so that then if (a + A, B + μ) are to form an orthogonal pair, the equation must be identically satisfied. Since (1) and (2) are true, (3) takes the form If ha, ha, ha, hμ represent the values of the gradients of a, ß, λ, μ, and if the angle at any point between the directions in which λ and ẞ increase most rapidly be denoted by [λ, ẞ], (4) becomes hx hp cos [λ, B] + ha hμ· cos [a, μ] = 0. (5) Whatever the sequence of the directions of the gradient vectors might be, the two angles which appear in (5) would be either equal or supplementary, and their cosines would be equal in absolute value, but the gradients themselves are intrinsically positive and the sequences must therefore be such that Suppose that in the case of two given pairs of orthogonal functions (a, B,) (, ), the necessary condition (6) is satisfied, and that the |