POINTS OF THE INTEGRAL CALCULUS. 



[135] 



Here is then a relation instituted between du and dv, which is forthwith tacitly 

 abandoned after being used. M. Cauchy makes the abandonment formal. He says (Moigno, 

 Vol. ii. p. 215.) " Des lors" that is after du has replaced dx, " ces deux variables devraient 

 etre independantes,'" for which no reason is given. Now it is plain that there is no objection 

 to our instituting a relation between du and dv in the element dudv. If we take u and v to 

 be new rectangular co-ordinates, and mark out the (uv) area corresponding to the (xy) area 

 over which we are to integrate, we may cover this area with elementary rectangles, in each of 

 which dv is a function of du, u, and v. We may not have a process of integration, inverse 

 to differentiation, by which to sum the results : but we want no such process for distinct concep- 

 tion either of the end to be gained, or of any approximate amount of acquisition. It is not 

 true, indeed, that there is a relation between du and dv in the final element above constructed : 

 the second dv is a different one from the first. But it is true that ff(x u y v - x v y u ) zdudv, 

 summed in the same order of elements as ffzdxdy, will not proceed first with respect to u 

 and then with respect to v, but in another mode, in which u and v vary together, both in the 

 first integration, and in the second. 



In the plane of my draw perpendicular straight lines having the equations w = const., 

 y = const. ; and in the plane of uv draw the curves which answer to these straight lines, 



x and y being each a function both of u and v. So that, if x = a in the first system, and 

 x = <p (u, v), then <f> (u, v) = a is the curve answering to the line x = a. Taking the 

 element 1345, the du of the transformation is (12) throughout ; the first dv is (23) which 

 is related to (12) by the equation a?„(12) + #„ (23) = 0; the second and final dv is (14). The 

 integral, first with respect to x, and then y, of ffxdx dy, if perfectly imitated in the trans- 

 formed integral, would require two integrations in which both u and v vary at every step. 

 And it may be noticed that as (1345) only differs from (24) by a quantity of the third order, 

 the usual mode of proceeding with fzdxdy may be imitated with the curvilinear elements. 

 But it is not intended that the transformed integral ffwdudv should be summed in this 

 way : a different subsequent resolution of the whole (uv) area, by lines parallel to the axes 

 of u and v, is always contemplated, and the absence of all notice of this step has, I dare say, 

 made the process as unintelligible to some others as it was to me. 



I need not enter upon the way in which the preceding remarks may be made to apply 

 to triple, &c, integrals. 



A. DE MORGAN. 



University College, London. 

 January 13, 1851. 



