OF ARTS AND SCIENCES. 237 



fi K . COS kIO — - j ^ K- . sin K (6 — - 1 

 - ^TTl - and ^;^+i 



in the case of the second motor. The second motor gives at everv 

 point velocity components which the first motor gives at a point 

 equally distant from the origin, but differing in vectorial angle from 



the first point by - . The second motor is then equivalent to the first 



turned through the angle - . 



If, then, the simple motor which corresponds to the function ~ were 

 transferred parallel to itself to the point (a, h) and then rotated 

 counter-clockwise through the angle -, it would correspond to the 



function ^ -, , .^-, . It is to be noticed that a rotation of 



the motor corresponding to — through the angle — - would by 

 reason of symmetry give the same motor, and multii^lying — by i 



would turn the symbol representing the motor through half the angle 

 which corresponds to a black sector.* The rational algebraic proper 



f («) 

 fraction -p.~. corresponds then, in general, to a distribution of vari- 



. . f(z) 



ously oriented simple multiplets, and the function A z + -rr ^ \ > which 



frequently appears in two dimensional problems in magnetism, to 

 a distribution of such multiplets in a uniform field. t 



16. Ihe real parts of the functions , — , are the ve- 



locity potential functions which correspond to two simple multiplets 

 of the Kth order, L and J/, of strengths respectively equal to X and /x. 

 Both L and M are situated at the origin. X's axis coincides with the 

 axis of X, but Jf s axis makes with the axis of x the angle a. 

 If we superpose L upon M, we get 



_ — {X -\- fi . cos K a -\- i . (i . sin *c a) 



= -'Li^^, (34) 



* This explains the identity of equations (28) and (30). 



t See, for instance. Maxwell's Treatise on Electricity and Magnetism, Vol. II. 

 Figs. XV. and XVI. 



