268 



DR JAS. BURGESS ON 



We luave here the coefficients in the following; values- 



A x = a + b + c + d + c+f+g + h + i + lc + etc. 



A 2 = 2&+6c + 14d + 30e + 62/+126<7 + 25a + 510i+1022;fc+ etc. 



A 3 = 6e+36d + 150e+540/+1806<7 + 5796&+18150i+55980&, etc. 



A 4 = 24tf+240e+1560/+8400£ + 40824&+ 186 480z+ 818 520ft, etc. 



A 5 = 120e + 1800/+ 16800# + 126 000& + 834 120i+ 5103 000/fc, etc. 



A 6 =720/+ 15120^+ 191 520A+1905 120i+16 435 440&, etc. 



A r = 5040*7 + 141 120A+ 2328 480i+ 29 635 200&, etc. 



A 8 = 40320/^ + 1451 520i + 30 240 000&, etc. 



A 9 =362 880i+16 329 600&, etc. 



A 10 = 3628 800&, etc 



K22) 



If we write A, B, C, etc., for the first terms of each value in the above, and reverse 

 the arrangement, we have — 



A in = iT+ etc. 



no 



9 



A Q =/+ „K, etc 



-JO 



A 8 =11+41+ ~ K, etc. 

 8 3 . 



A 7 



G+ i ff+ i2 I+ ir K ' 



A K =F+3G + 



77 



ii 



19 



H+ni+W-r. 



10 



2fi 



240 



331 T , 45 



^= D + , E +^F+lG + lH+^I+^K, 



\ (23) 



43 



23 



•.-°+i»+i*+?+m°+m* + 



605 T 311 K 

 12096 20160 ' 



*>= B+c+ r2 B+ l M +m F+ m G+ mzo R+ i 



17 I+-™-K 

 259200 ' 



. , , 1 p , 1 „ , !n, E , F L # > H . J . K 



A, =A + T2 B + l C + -D+ ^ +_++__+_+_ 



15. These equations readily give us the values of A, B, C . . . K ; and now, if 

 n denote any subdivision of the intervals for which A, A 2, A3, etc., represent the 

 successive differences, and A', A^ A3, etc., represent the differences for these smaller 

 intervals in the value of the argument, — then we have — 



n 10 A\ n = K+ etc. 



ID 



n?A' 



2n 



etc. n s A' R =H-\ + -j-- r-; and so on. (24) 



n 3ii- 



the index of A in the second line, gives the value in the m+1 column : thus A 3 7 + A 4 7 = 1806 + 8400 =10206, ami 

 10206 x 4 = 40824 = A 4 e . The formula is derived from that for A"0m in Hersciikl's Appendix to Lacroix'.s Differ, 

 and Intecj. Calculus, (1816), p. 478. 



