2(50 DR JAS. BURGESS ON 



distance apart is = ty- t Q} \ = V, - V , A£ = V n - V B _x , and A 2 , A^, the first and last 

 of the second differences, and so on ; then between V n and V„ the area is — 



e{(gV„ + v, + v, + . . + jv.) - 1(a; - aj - ^ a; + a 2 ) - ^ a; - a 3 ) - Ija; + a 4 ) 



863 .., . . 275 /A/ A , 33 953 /A , . . 8183 , A , A x 

 " 6T480 ( A * " A ') - 24l9 2 ( A « + A «) " 362T806( A ^ " ^ " 103080()( A * + A °) 



3250 433 

 479 001 600" 



(A 9 -A 9 )...}* . (17. 



of which expression the first three terms will generally be sufficient. Taking the same 

 example, we have — 



A,= -681145 A 2 = +994 £V + V, + ..+ V a + iV 10 = '290 419 6916 



A;= - 672 183 A' 2 = +997 - T L(A,-A,) = -747- 



A;-A i= +8962 A; + A 2 =+1991 -^(A; + A 2 )= - 8 3- 



Sum, . . -290 419 6086 + 



and 0= -01 ; hence the area is '002 904 196 086 + , as before. 



For a single interval, as between V and V x , by putting A° for the second difference, 

 derived from Y_ l and V x , and A 2 the next in succession, derived from V and V 2 ; A° the 

 fourth difference, in line -with V , and A i5 for the next below, etc., we have the area 

 expressed by — 



O(V. + V0-^(AS + A0-^(AJ+A0- I ^ o (AS + A.)- y ^ 5 (A« + A a ). . . { i«) 



Taking the values of Vat 1*130, 1-140,.. . and 1 "180, we find for ri60, A° = +99 373. 

 and A 2 = +99 759, also A°= +70 and A 4 = +66. Then — 



£(•293 811 239 + -287 044 575) = 290 427 907 



-5^(99 373 + 99 759)= -8297 + 



-T^o(70 + 66)= - 1 _ 



Sum, as before, nearly . . -290 419 609 — . 



Interpolation. 



13. The method of interpolation employed is familiar, but the process may be 

 explained by which the transference is made from the differences found from the 

 computed values, to the differences required for those to be interpolated. t I have not 

 met with it in any text book at my command, and I think the formation of these 

 differences indicates that too much stress may be laid on the common warning that most 

 reliance is to be placed on results which lie nearest the middle of the series of values 



* Conf. Db Morgan's Diff. and Integ. Calc., pp. 262, 313-318 ; Woolhouse, Assurance Mag., vol. xi (1864), p, 809. 

 By tins method the computation might have been abridged in some portions, had I noticed its advantages earlier. 



t Mr W. T. B. Woolhoube, in a paper " On Interpolation, Summation, and the Adjustment of Numerical 

 Tables," in The Assurance Magazine, 1863-65 (vol. xi, pp. 61-88, 301-332, and vol. xii, pp. 136-176), has developed a 

 formula with necessary -tables for interpolating terms in the middle interval of a series. The treatment is interesting, 

 and the formulje are rapidly convergent, but not altogether convenient for computing a lengthy table. 



