Field oj Circular Currents, 383 



whence we find, by expansion and the known relation 



P k* 21 7fi 

 ^ == ll5- f 32 + 1024 / ' + -' 



the following powers of: q expressed in terms of the co- 

 ordinates. As they are of great convenience in various 

 calculations of similar nature, it will not be out of place to 

 give them to the 7th power of f and rj. 



q la^Ua* 4a 3 "*" 512a 5 16 a 5 "^ 4a 5 2048a 7 



1 



309gy 9|V_|^ 6 

 512a 7 + 8a 7 4a 7 + 



Y (14) 



' y 1 6a 2 + 32a 4 8a 4 + 1024a 11 4a 6 + 16a 6 + 



3 _ J* 3f 3gV 75g 7 27gy 3gV __ 

 ? ~ 64a 3 "*" 256a 5 64a5 + 8192a 7 256a 7 "*" 32a 7 ■ " 



9 ~ 256a 4 ^ 256a 6 64a 6 ^ '"" 



.- P , H 7 sgy 



* - 1024a 5 4096a 7 1024a 7 t 

 2 ~ 4096a 6 ^ "•' 

 ?7= f6384a 7+ •■" 



v/ - _ i A /n 1 . p _ _f_ + isp. . jggy ,v 



V? "2Va| ^Sa 2 2a 2 ^ 256a 4 16a 4 "^a 4 

 1607_5f 551 gy llogy 5t7 6 

 + 2048a 6 + 512a 6 + 64a 6 16a 6 " 



From these expressions we easily find, by substituting 

 in the expansion in terms of a, the following formulae 

 for X and Y : 



T T e Jo . *5P 15^,525? 5250y 4 _lO57 7 4 ]. \v 



Y - * /? , 3 ? 2 o^? 2 , *5F_4ggY 15^' 6175ft 



" a 1 " t "2a 2 a 2 32a 4 4 a 4 + 4 a 4 128 a 6 



1575 |V 525 £V- 35 ^ 6 . \ 



"64" a 6 + l6 a 6 ""8 ? + "' J" / l j 



