168 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where ^,„ gi, g^, g% are given positive constants, and let the value ( U^ 

 of U on the cylindrical surfaces be a function of z only (the same for 

 all the surfaces), such that 



Z7^ + 





where l\, h, h, h are given positive constants. Then if U has the 

 constant value ^o at all points in so much of the xi/ plane as lies 

 within Sq and the value zero at all points on and within /S', for which z 

 is positively infinite, IT is determined in the positive space within Sq. 

 For if we assume that there could be two such functions and apply 

 (35) to their difference («) in each of the regions r,,, ti, tj, Tg, • • • , 

 multiply the resultant equations by k^^, ki, k-i, h, ■ ■ ■ , and add them 

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

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

 .0:1/ plane. 



It is to be remembered that 



??+?? (47) 



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), 

 ^'i, ^'2, ^3, • ■ • , are all equal, ^1, g-^, g^, ■ • ■ , are all equal, and all the 

 w^ areas ^4i, A^, Az, ■ ■ ■, are alike in form, however they may be 

 oriented. In the region t^, U is everywhere equal to Cs, which is, as 

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



^H-..f^..|X/-(|f.f)u. («) 



where / and k are given positive constants. 



If in this case we find for every one (t,„) of the regions ti, t^, T3, • • • , 

 the function (Wm), which within (r,„) satisfies the equation 



dir,„ fd'w„, 8hv,n\ .,,,. 



dz ^' V 3.r ' dr 



