Journal . 1. IF the inclined lines in the figure represent the products ofthe quantities they connect, so that AB=iioX ^^o; then A F — AB = (w, — Mq) Aro=Awo {i\ — ro) = CD — ED AB + CD = AF + ED Adopt the convention that a line taken rightwards gives the-\-ve sign to the product, and leftwards the —ve sign; so thatAB = UqX Ai-o, BA= —KoX Aro. then AB-fCD + DE + FA = 0 Therefore, if we regard the horizontal lines as valueless, we maysay that the circuit of the cell is zero. It will be noticed thatthe relative positions of Vq, Vi, Avo, are inverted and reversedas compared with the relative posi
Journal . 1. IF the inclined lines in the figure represent the products ofthe quantities they connect, so that AB=iioX ^^o; then A F — AB = (w, — Mq) Aro=Awo {i\ — ro) = CD — ED AB + CD = AF + ED Adopt the convention that a line taken rightwards gives the-\-ve sign to the product, and leftwards the —ve sign; so thatAB = UqX Ai-o, BA= —KoX Aro. then AB-fCD + DE + FA = 0 Therefore, if we regard the horizontal lines as valueless, we maysay that the circuit of the cell is zero. It will be noticed thatthe relative positions of Vq, Vi, Avo, are inverted and reversedas compared with the relative positions of Uq, Wj, Amq- 236 Actuarial Notes. [April (n + 2 (ll+Z\: ( + l)o («-l) (n-2). 2. The 2nd ligure is the ordinary scheme of differenceswith binominal coefficients inserted^ the symbol (n),. being used for n{n — l)(n — 2) . . (;* — ?• +1) The coefficient {n)i is placed opposite Wq and the law followed bythe remaining coefficients is obvious. The coefficients constitute a scheme of differences invertedand reversed—thus (71 + 2)4—( + 1)4= (* +1)3—and each cellhas the same properties as that in the first figure. Consider any group of cells. The circuit of each cell iszero^ and therefore the sum of the circuits of all the cells is the products represented by internal lines all cancel out,since each is taken once positively and once negatively. Thereremain only the external lines, and therefore the circuit of anygroup of cells is zero. It follows that if we pass from any chosen coefficient, as (n)i,to any chosen difference, as A*m_2, the result is identically thesame, whatever route be taken. For example (n)lAt<_l+(n + l)oA??^M_l + (?^ +
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