TEANSACTIONS OF THE SECTIONS. 7 



the theory of numbers. The author's attention was first directed to this connexion 

 by Professor H. J. S. Smith, of Oxford. 



On a Method of discovering Remainders in Arithmetical Division. 

 By C. M. Ingleby, LL.D. 

 Let A be any number (or aggregation of units), and let the prefix R indicate 

 the operation of distributing the number to which it is prefixed in a scale of nota- 

 tion whose radix is r. Let B be the number of units in B A ; then, par iter ME is 

 the operation of distributing B in a scale whose radix is in ; and so of other prefixes. 



Then is an integer, m being + or -. Now, if MB> r, let C be the 



/• — m 



Tt V — MC 



number of units in it. Then — — is an integer; and so on until we arrive at 



r—m ° 



-p a TYTT 1 "R A Ml 1 



MT, a number <r. Then, finally, — is an integer ; — — and have 



■" r — m r — m r—m 



equal remainders, and as MT<r, if+, it is either = r — m, or is the remainder after 

 dividing RA by r—m ; and if MT is — , r-m-MT is the remainder sought. 



Apply this to the denary number 76438 ; then r=10. 



(.1) Let »sl, then the divisor is 9; 



MA=8+3 + 4+6 + 7=28; MB=S+2 = 10; MC = 1, 

 the remainder after dividing 70438 by 9. {This is an extension of the principle of 

 "the ride for casting out the nines.") 



(2) Let m= — 1 ; then the divisor is 11 ; 



MA=8-3+4-6 + 7 = 10; MB=-1; and 11-1, 

 or 10, is the remainder after dividing 76438 by 11. 



(3) Let m= — 6 ; then the divisor is 16 ; 



MA=8-3x6+4x36-6x216+7x 1296=7910; 



MB = -6+9x36-7x216= -1194; 



MC = -4+9x6-36+216=230; 



MD= -3x6+2x36= 54; 



ME = 4-5x6=-26; 



MF=-6+2x6=6, 



the remainder after dividing 76438 by 16. 



The theorems dependent on this general principle are as follows :— Writing SA 



for the ultimate operation on A instead of MT, and letting A w A 2 , A 3 , 



Ap be /i numbers, 



' S^.A.-A, A fl ) = S(SA,.SA 2 .SA 3 SA M ) (1) 



Cor. If A 1= A 2 =&c =A W SA^=S(SA) M (2) 



Also S (A 1 +A 2 +A 3 + ...... +A„) = S (SA 1 + SA 3 +SA 3 + +SA M ) . . (3) 



Cor. S{K 1 + K 2 +K"+----+^)' 1 



= S[(SA 1 )"'< + (SA 2 ) m 2+(SA 3 )'"'+ +(SA^) VJ». . . (4) 



P.S. If m be + and > 1, and MT (or SA) > m, (it must be < r), we have appa- 

 rently a case of failure. 



On a New Method in Geometry. By Prof. Plucker. 



On the Extension of Taylor's Theorem by the Method of Derivations. 



By Prof. Price. 



On some Applications of the Theory of Probabilities. By Prof. Price. 



