Miscellaneous Papers. 257 



set of that series ; and the sum of the several differences taken 

 equals the highest term of that series. 



We may put these principles into intelligible working for- 

 mulae by adopting symbols for the several quantities expressed. 

 For the first, let a, b, c, etc., to p, q, represent the several terms 

 of a series, p and q being the last two ; let n be the number of 

 terms in the series and let S be the total sum. Then a general 

 formula will be : 



S = a?i + (6— a) (w— 1) + (c— 6) (7i— 2) + (d— c) (w— 3) . . . 

 {q—p) (n— (n—l). 



Expanding, 



S = an + bn — an — b + a -\- en — bn — 2c + 26 + dn — en — 3d 



+ 3c . . . + q. 



Canceling and collecting, 



S = a-\-b-\-c + d . . . -\- q. 



q. e. d. 

 Again, in a series for a magic square, when it becomes nec- 

 essary to write the series in n sets of n terms each, let Aa, Ah, 

 Ac, to Aq, represent the first set; let Ba, Bb, Be, etc., represent 

 the second set; Ca, Cb, Cc, etc., represent the third set, and so 

 on ; let n be the number of terms in a line and S equal the sum 

 of the numbers in a line ; then the formula will be : 



S = Sn = Aan+(Ab-Aa)(n-l) + {Ac-Ab){n-2)... + iAq-Ap)in-(n-l) + 

 Ban+iBb-Ba){n-l) + {Bc-Bb){n-2)... + {Bq-Bp) {n-(n-l) + 

 Can+{Cb - Ca) (n- 1) + (Cc - Cb) (n-2). . . + iCq- Cp) (n- (n-1) + 



Qan+iQb-Qa)\n-ij + {Qc-Q^^^^ 



Expanding, the equation will be : 



S = Aan + Bbn — Aan — Bb + Aa + Ccn — Bbn — 2Cc + 256 

 . . . + Qq. 



Simplifying, 



S = Aa -\- Bb + Cc . . . + Qq, 



which is a self-evident fact. 

 An apparently simpler formula, though not so explicit unless 

 well understood, would be to let d°, d', d", etc., to d°, represent 

 the differences between the successive terms of any set (d° be- 

 ing the first term or the difference between itself and zero ; and 

 let D', D", etc., to D", represent the differences between the 

 initials of the several sets ; then the formula will be : 



S = d°n± (D' + d') (n-1) ± iD"+d") (n-2) . . . ±(D^+d^) (n- (n-l). 



Nothing, however, is gained by this formula as it is irre- 

 ducible. It must be known and expressed in the formula 

 whether any term is greater or less than the preceding term, 



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