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, 190G. 



If <^i (.r, I/), (p2 (•'', ]/) are the potential functions due to two colum- 

 nar distributions of matter the lines of which are perpendicular to 

 the a: y plane, and if ij/i (.r, y), xp^ {x,y) are conjugate to <^i and ^2, 

 respectively, the families of curves obtained by equating \pi and 1/^2 to 

 parameters, are lines of force of the two distributions. Moreover, 

 ^1 + 02 is the potential function due to a combination of the two 

 distributions, and the function i/^i + ^2 equated to a parameter gives 

 the corresponding lines of force. The fact that if (<^i, i/^i) are any 

 pair of conjugate functions and (02, ^^ any other such pair, the 

 functions {a 4>i + h 02> ^^ "Ai + ^^ "As) 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 combi- 

 nations of simple elements. If (0i, \pi) are merely a pair of orthogonal 

 functions and (02, ^i) another such pair, it is generally not true that 

 (01 + 02, "Ai + "As) are an orthogonal pair : thus {x, y), {x^ -f //-, yl x) 

 are pairs of orthogonal functions, but x -{■ x"^ ■\- y^, y -^ D /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- 



^ 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. 



