368 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



of <£ will be continuous, and the latter will satisfy the equation y/ 2 <£ = 0. Denote by 

 dT the element of the region T; then 



dT= dxydx^iXi <h' n = r"' 1 ( II sin a da k ) d<f>"-\h: (20) 



The element of the boundary S is dS, and this is readily found to be given by 



dS = B" ~ x ( ' TT'sin" a'~ \ld k ) dot, _ a (21) 



(B, a\, a 2 , . . . . o' n _! being the limits of r, a u o^, . . . . a n _ 1 ) 

 Introducing now an angle 8 defined by 



it = n — 1 j = k k = n — 1 



cos 8 = cos a x cos aj + 2 {cos a t + 1 cos a k + l IT sin a,- sin aj} + II sin a fc sin a£ (22) 



*.• = i j — i it = l 



The introduction of this angle 8 enables us to obtain quite easily 



»=%/■■■■/• *<*-->" , (23) 



2^5 *f '/ ^i B (i? 2 + r- - 2 Br cos 8)* 



^ denoting the arbitrarily chosen surface value of <j>. 



Further consideration of this function <f> will be deferred for the present, as a full 

 investigation of its properties would require the development of the theory of 

 spherical harmonics for a space of » dimensions. Equation (17) will afford the means 

 of obtaining some of the more elementary of these properties. 



We will resume here the consideration of the quantities g jk . Observe that if we 

 denote the quantities of the left-hand sides of equations (6) by a single letter, say by 

 7] U ifa, . . . . that we have identically 



rf^ + rfc + •••• + ^ j ~ 0- W 



If now from the t/s we form quantities similar to the f's and call them £$, £$ .... 

 we shall have a set of equations of the same form as (6) with f replaced by £ (1) and 

 u replaced by rj. Calling the left-hand members of this new set rfi\ r$\ . . . ., and 

 again performing the same operations, we get equations similar to (6) with the £ 

 replaced by | (2) . and for the right-hand members y* 3 u, \7 22 u 2 , etc., where y 22 means 

 that the operation y 2 has been performed twice. This process can of course go on 

 indefinitely or until we come to such values of the ^-functions that 



rj^dx, + rtfdxt +.... + -q^dXi 



is an exact differential. Instead of treating these functions u l ,u 2 - . . . U{ altogether 



