346 EEPOET — 1880. 



1. Simple measurement a; :=— (6a) 



2. Simpson's rule jy dx =— h{a + 4& + c) 



3. Woolley's rule / / z dx dy =— - hh (aa + &i + 2^2 + ^3 + C2) 



4. Merrifield's Hlfy dx dy dz =^ liU {a^' + 5, + &/ + l^' + h^" + C2') 



or, as the multipliers may be graphically arranged, 



1 11 



6; 141; 121; 101; 

 ] 11 



there being a curious tendency of the middle ordinate to ' move out.' 

 The late H. J. Purkiss, by operating upon the equivalent form 



Wo = «o + ^*2 +_ By2 + C^2 + 



(w variables)* 

 showed that the n"' integral 



'^-f-nf-kf-i ""^""^y^' 



was represented by 



V = 1 2'"' hid I S - 2 (n - 3) Wo I 



when Uq =/ (0,0,0 ) and 



2:=/(7.,0,0....)+/ (-7^,0,0....) 

 + /(0,7<;,0....)+/(0,/^-^,0....) 

 + f (0,0,1 ....)+ f (0,0,-1, ....) 

 + &c. 

 Multiple integrals of the form 



///■ 



u (dx)"' = U„ 



may be treated by rules nearly similar to those already given for simple 

 integrals. It is worth while to observe that in this form of integral only 

 one integration (the last) is between definite limits, the others being 

 rather algebraical forms expressed by the notation of the integral calculus, 



than actual integrations. Thus 1 1 u dx^ loetween the limits ±11 is not 



the analogue of / / m dx dy, where 1i and Z; are each made ^ n, 



J -i^J -k 



but is an abbreviated expression for / dx f u dx. This becomes evl- 



J -n Jo 



c7"'U 

 dent, when it is remembered that U„. is simply the solution of ~y^" = u. 



Moreover, after one integral has been taken between limits, all following 

 integrals are mere multiples, with a parabolic series added, on account of 



* See Trans. 1. N. A., vol. vi. for 1865, p. 48. 



