ON APPROXIMATE MULTIPLE INTEGRATION BETWEEN LIMITS BY SUMMATION. 605 



DOUBLE INTEGRATION. 



in 



There is a particular kind of double integration which can be treated 

 a somewhat similar manner, namely, when the quantity to be integrated is a 

 function of a linear function of the two independent variables. 



Thus if I=t'\udxdy ............ ....... ..m 



J *i J yi 

 where u is a function of r, and 



r = a + bx + cy .................................. (2), 



let x = x x -x 1 ) ........... ...... .......... (3), 



!&) ........................... (4), 



-. .................. (5). 



If we write 



t + y l ) ........................ (6), 



. ..................... (7), 



................. . ............ (8), 



............ .......... (9), 



we may consider u as a function of of the form 



. ..... v ........ (10), 



Now let be the value of u corresponding to = 0, 



tt, and u\ C= 4. 



u, and < = 4> 



and if we assume 



! = (**- ,) (ft - ft) Rw + A (i + ',) + R, (u, + u',) + &c.} ...... (12), 



