194 Dr. L. N. G. Filoii on a New Mode of 



2. Let us assume that / is so expanded. Then Y is ex- 

 pressed as the sum o£ typical integrals as follows : — 



V = I /„(<2; cos V + ij sin r + iz)dv + 



r, 



+ ) /"' {^ ^^^ ^" +3/ ^ii"^ ^' + ^'^) <^os mv dv + . 

 » 



-|- 1 F„j {x cos t* + y sin v + iz) sin ??ii' ^/r + 



'0 



the f's and F's being all analytic functions over the range 

 of x^ y, z, V considered. 



Transform to cylindrical coordinates 



.r = p cos (^. y — p sin </>. 

 1 /.^ (<2^ COS i^ + 2/ sin V + 24:) cos mr dv 



1 



~ 1 fm{p cos (z;— (/)) + ?;:r) cos mv dv. 



Jo 



Let v — ^ = u. 



fm (p cos?^ + ?r)(cos ???(/) cos??i?« — sin ??i<^sin?/iw)c?i/. 

 Consider 



cos ??20 i fm {p COS w + 2c) COS mil du 



= cosm^ I -f- I ]. 



ko J-?-J 



Li the second integral write n' = '2'Tr + u, 



I /m (/O COS Zf + IZ) cos ?72Zi <:Zi« 



J-? 



= ) fmipcosu^ -\-i:) COS mi/ du', m being an integer. 



2jr-(P 



I. 



Hence the two integrals together 



/^27r 



= COS mcf) i /„, {p cos z^ + iz) cos ?>2Z< fZn 

 Jo 

 Similarly^ 



1 /«! (/o COS u + ?c) sin mu du 



J-(P 



= 1 fm{p COS z« 4- 2^ sin mu du 



= 1 /m {pGOSU-\-iz) sin mudu+ 1 /^(/ocos /; + ?2') sin mudu. 

 Jo ' J- 



