168 PROCEEDINGS OF THE AMERICAN ACADEMY*. 



when' g 0l <i u g%, g» are given positive constants, and let the value ( /' 

 of £7 on the cylindrical surfaces be a function of z only (the .same for 

 all the surf te . Juoh that 



+ ^/7'(- ;+ -') / ' l!+ '•• =,, ' , "" 



where /,„ fa, fa, fa are given positive constants. Then if U has the 

 constant value /', at all points in so much of the xy plane as lies 

 within Sq and the value zero at all points on and within S for which : 

 is positively infinite, U is determined in the positive space within S 

 I'm- if we assume that there could be two such functions and apply 

 (35) to their difference (m) in each of the regions r , t x , t 2 , t 3 , • • ■ , 

 multiply the resultant equations by /,,. fa, k%, fa, • • •. and add them 

 together, it will be easy — to show in the way indicated under (II) 

 — that u is zero everywhere inside S on the positive side of the 

 try plane. 



It is to be remembered that 



c-f d*U 



■jj + -5— - (4 1 ) 



c.r ay- 



is an invariant of a transformation of orthogonal Cartesian co-ordinates 

 in the xy plane. 



(V) In an important special case similar to that stated in (IV), 

 fa, fa, fa, • ■ ', are all equal, </i, g%, <i,-„ • ■ •, are all equal, and all the 

 ir areas J,, .1., J .„ • • • , are alike in form, however they may be 

 oriented. In the region r , V is everywhere equal to U a , which is, as 

 before, a function of z only, and the surface condition becomes 



where / and k are given ])ositive constants. 



If in this case we find for every one (r m ) of the regions t u t 2 , t 3 , • • • , 

 the function ("•„,), which within (t m ) satisfies the equation 



' ' f r1 "' , c ~"'">\ /,o^ 



' : V Ci j J 



