﻿VII. 



On some Applications of the Method of Mechanical Quadratures. 

 By GEORGE P. BOND. 



{Communicated to the. Academy, May 29th, 1849.) 



It is proposed, in this communication, to apply some of the known formulae of the 

 method of quadratures to several astronomical prohlems of interest, where, under 

 certain conditions, their employment tends to accuracy and simplicity in computation, 

 and offers methods entirely independent of those in common use. 



I. Let A v , A\ A a , be a series of values of any function of which / is the inde- 

 pendent variable. For convenience of expression, we may call t the time for which 

 A a , A x A„, are given. Supposing A to be computed numerically for the succes- 

 sive values of t, t — 0, t = x, t = 2x t = n r, x being the equal interval between 



each succeeding value of t, and n the number of times for which A is found. We may 

 then arrange t and A with its first, second, &c. differences, J\ J' 2 , J 3 , &c, in vertical 

 columns, as follows : — 



(=0 A, 



A 



t = lr A i A 



j\ A 



t = 2i At J\ 4\ 0) 



A A 



t = 3z A 3 A 



A 



t = 4t A, 



A A 

 It is plain that, as x is decreased, — °, — ', &c, will approximate to the values of the 



first differential coefficients of A at the times t = ix, t=lx, &c, which they will ac- 

 curately represent as long as the third differences of A, or J\, J\, &.c, arc insensible. 

 If, then, we know A , and also the first differential coefficients of A for t = $x, J = 1 r, 

 &c, these multiplied by x, and substituted for J l , J\ J]„ in (1), will give, by 



