208 



Again, 

 (rt, b, . . 



(V ViI 

 (VV 1 



(a n _i/3 n _i) 

 V Vxlan-i 



where (y Vi vl 2 &I X 1^ A) = (V Vi vl 2 / 



It is further shown that the effect of the operations v, v l on a 

 given function, u=*Zaix n - i y i , may be represented by 



and the case of 

 The value of 



is examined in detail. 



in terms of *, - , * 2 , . . * t -r-, * 2 , . . are calculated, (1) when 

 ax ax ay dy 



s lt * 2 , . . are any linear functions of x, y, (2) when they are any 

 functions whatever; and, in case (1), the effect of the above opera- 

 tion on a given function is determined. 



IV. " Problem on the Divisibility of Numbers." By FRANCIS 

 ELEFANTI, Esq. Communicated by ARTHUR CAYLEY, Esq. 

 Received November 24, 1859. 



Problem. To find a proceeding by which the divisibility of a 

 proposed integer N by 7 or 13, or by both 7 and 13, may be deter- 

 mined through the same rule. 



Solution. We can designate the number N by abed . . . mn, so 

 that (a) be the first or highest, and (n) the last or lowest digit in 

 it, therefore we may put 



