SURFACES IN HYPERSPACE. 



293 



and the last bracket becomes a single term repeated. Moreover the 

 first bracket may be changed by interchanging the indices i, j, Ic. For, 



^ diaij) d(aik) 



dijk oy,- 



since in either case the summation is over all values of _;" and k. Hence, 



dXgdXr 



This somewhat cumbrous form may be simplified by introducing 

 the notation of the Christoffel symbols. 



(32) 



The above expression then becomes 



sr 



t 



= '^iiklrilsi'Ytk 



k 



OXadXr 



To complete the solution for the second derivatives, multiply by Cti and 

 sum over t. Then 



'^tCtl 



r s 

 t 



= 'Eijyrr/sj 



I 



+ 2i(au) 



d'yi 



dXsdXr 



Next multiply by (a*™'^) and sum over /. Then 



2,;c,,(a(-0) 



r .s 

 t 



^mlrr^siia^"''') 



'if 

 . I . 



+ 



d^ym 

 dXrdx, ' 



and hence finally we have the expression 



d^ym 



dXrdXa 



:^ticti{a^"''^) 



r s 

 t 



- ^iiUrrYsiia^'"'^) 



^3 



I 



(33) 



