42 EEV. T. P. KIRKMAN ON MNEMONIC AIDS 



(A.) If e and m are mups, mutnsMy jsrimes, and m is prime, 

 vkm, (8, g) or e : m, has the remainder of e' vi, or e"" ~* (e : m), 

 or e^tm; 1 is m —< 1, as in (7, a). To prove this, i/ou put 

 e" rz (1 + d)", Sum to power wi ; Mn is unity : rf is e — 1, 

 (7, a) : then put rf^ =: (1 + c)% or (1 + (e —2))'°, c being 

 d, — 1, (7, a)^ The remainder of e" : ra is thus seen to be that 

 of {1 -[- (e — l)"") : m, also that of {2 + (e — 2)"} : m, &c. 

 The two extreme terms of the expansion tcill have the remainder. 



(B.) If g, being less than m — 1, (1 over g), is such that e*' is 

 em X integer, or e*"" is divisible by m, then I is gy^ int, or 

 m — 1 is divisible by g ; and if then e is no such number as g^ e is 

 one of the ^rtwitive roo^s ofm^ Vide Murphy's " Equations," 

 §61. 



Let not the rhymes and scanning provoke a smile. There is 

 some science, mnemonic and mathematical, in the rhyme ; and 

 the ear is ever grateful for the humblest jingle. It is a hundred 

 to one that the reader is continually indebted, to this day, to 

 an old and not very neat mnemonic, " Thirty days hath Sep- 

 tember," memorable for its rhyme : 



" Except in leap year, then's the time, 

 February's days. are twenty and nine.' 



(18.) As examples of series and expansions, the following 

 may be introduced ; and first the binomial theorem 



Suton (un 'r) ? — write fn* on r ; (pron. Frow.) 

 (A.) Then r to i you multiply 



with (i-n-backs by i fags.) vid. (11, c, a) 



(B.) If 'n has den e, put 'r vi (re) ; num dits wed e. 

 (A.) Suton un r is s^cm to power n of (unxij and r.) What 

 is the expansion of (1 -|- r)° ? Write fir" on r ; i. e., 



ro-l- ri+ r^4. . . . 



on means with rising indices j b on is b -j- b^ -j- b^ -f- • • • 5 

 h fr o», is b** + b^ + ^'^ + * * ' J or b from «Je» power on, 

 b on from zero power. Then r to i or r* you multiply with (or by 

 (n-n^^^ • dT-^ . . • n — i — . 1 ) : l'2-3""i. 



