Kellehek — A 3 -Dimensional Complex Variable. 87 



satisfy the conditions (A) are themselves expressions which satisfy the 

 same conditions. Similar relations hold when we deal with expressions 

 P, + P 2 + . . . P„ 0"" 1 , where 0" = 1 but 1 + + 2 + . . . 0" 1 is not zero, 

 and P u P 2 , . . . P n are functions of n independent variables x,, x~, . . . x n . 

 The differential coefficient exists provided 



dP,„ _ dP m+li 

 dz,,. dx^k 



for all values of m, n, and k from one to n, n being subtracted from the suffix 

 when the suffix exceeds n. 

 Likewise, if 



(P, + P,0 + . . . P n 0"" 1 )* = i\' + P:e + . . . P n ' 0"", 

 and if 



dP m _ dPrn+k 



dx^ dx^k 

 for all values of m, ft, k as above, then 



dP ,„ _ dP mtk 

 dx» dx^ k 



III. It follows from II that, since 



u or x-i + z 2 + . . . x„ 0"" 1 



satisfies the conditions 



dP m _ dP„u.k 



dx^ dx^ k ' 

 any expression which can be formed with powers of u, positive or negative, 

 integral or fractional, will be such that when written in the form 



P, + P 2 + . . . P„ 6"\ 



the relations 



dP,„ _ dP m+k 

 dxp. dx^k 

 will be satisfied. 



Hence, when we consider the 3-dirnensional variable, it follows that, 

 when fix + yO + zQ-) is written in the form P + QO + B.Q-, and x, y, z are 

 the distances of a point from three rectangular planes, the normals to the 

 three surfaces P = const., Q = const., P = const., which pass through a point, 

 make equal angles with one another and with the line x = y = z. For the 

 cosine of the angle between the normals to P = const, and Q = const, is 



dPdQ dPdQ dP dQ 

 dx dx dy dy dz dz 



JdPV (dP\* (dP\=n,dQ\r fdQ\r ,dQV-> 



[13*] 



