8o SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



quadrant in which x and r are positive, and hence deal only with 

 positive values of jj. and the radicals. 



In terms of the coordinates (X, fi) equation (i) becomes 



1 \(r-+i) ^^ 



+ ^ 



(.-.= ) |]=o. (4) 



as a straightforward change of variable will show.^ We follow the 

 usual method of integration and try for particular solutions of the 

 form ' 



a product of a function of ^ by a function of fi. Then (4) becomes 



I ^ 

 Z dC 



(r+i)f 



^ M d/i 



^'-"-^ d,\ = 



o. 



Here the variables are separated and the equation could not hold 

 identically unless the two parts were equal and opposite constants. 

 If we set 



^J^{i-r)'^ff\+n(n+i)M^o, (5) 



'^"^^(^ + 1) '^^] -n(n + i)Z = o, (6) 



we see that the first is Legendre's equation and the second a slight 

 modification of it. For n = o, 1,2,...., the polynomial solutions of 

 (5) are respectively constant multiples of i, fx, 3/i- — i, . . . . ; and of 



(6), I, C, 3^ + 1, 



A consideration of (2), or of the figure made up of the confocal 

 ellipses and hyperbolas, shows that for large values of ^, i. e., in the 

 distant portions of the plane, ct, = p and ^ = cos"^ /^ are approximately 

 polar coordinates with the ji'-axis as polar axis. Then in these regions 

 we have approximately 



^ — —sin ^ J^ = — (tangential velocity along p = const.). 

 pu(7 POjU 



^ It is easier to transform (4) into (i), and still easier to express directly 

 in terms of \ and n the condition of continuity; for if dse and dsh are ele- 

 ments of arc along the ellipses and hyperbolas, the velocities are — , 



and — .-— . and the flux is 

 dsh 



Q>S}i \ aSe I OSe \ Q)Sh I 



which is readily expressed in terms of ^, M- 

 ^ See, for example, Wilson's Advanced Calculus, Chapter XX. 



