[134] PROFESSOR DE MORGAN, ON SOME 



multiple integral. In my Differential Calculus, (pp. 3y2 — 395,) I abandoned this mode, and 

 substituted another, which for the case of two variables, as shewn, and of three, as might 

 be shewn, is perfectly sufficient and not very difficult. Having thus evaded the necessity 

 of considering the usual demonstration, I have never, until now, endeavoured fully to settle 

 in my own mind the question whether the doubts which I entertained about it were or were 

 not well grounded. This I have now done, and have satisfied myself that the fault is one of 

 omission only, and that the omission can be easily supplied. It would not become me to 

 conclude that analysts do not supply this omission, each for himself : and yet I am forced 

 to suspect it by observing that the most recent writers, as M. Cauchy and M. Catalan, (cited 

 by M. Moigno), neither mention omission, nor supply it. 



In making the summation expressed by ffz dx dy, any process is sufficient which 

 divides the whole extent (say an area), over which integration is made, into elements of the 

 second order, n in number, n being infinite, and sums all, or neglects a number of an order 

 lower than n. It is not necessary that to one value, or rather position and value, of dx, 

 should correspond an infinite number of cases of dy : it is enough that the whole area signified 

 by ffdx dy should be subdivided into an infinite number of rectangles, each of which has all 

 its sides coinciding with sides, or parts of sides, of others. We have then to shew that the 

 particular mode of treatment we propose for ffdx dy just, as it were, covers the ground: 

 and we have then to treat ffssdx dy by that mode, or another. There is an unlimited number 

 of ways of dividing a given area into rectangles of the sort of contiguity required : and it is 

 obviously allowable to shew one way of filling up the required extent, and afterwards to use 

 another way in summation ; provided always that we assure ourselves of the equivalence of 

 the two extents which are used. Now it is the sole defect of the usual proof, that in con- 

 verting ffzdxdy into ffwdudv, the mode of dividing into elements of the form du dv which 

 is adopted in the proof of equivalence, is not that which is intended to be used in the subse- 

 quent integration, and that this is not stated. Legendre's* proof, changing symbols, is lite- 

 rally as follows, (Mem. Acad. Sc. 1788, pp. 458, 459.) 



Let u and v be two variables which we are to introduce in place of x and y : suppose that 

 between these four quantities we have established two equations, from whence we have 



dx = x u du + x v dv, 

 dy = y u du + y v dv, 

 the value of dx in s&dxdy must always suppose y constant ; so that for substitution we have 

 y u du + y v dv = 0, or dv = — y u du + y v : whence 



dx = W « y *~' V ° y " du, xdxdy = a, « % ~ iV ° y " zdudy. 

 Vv V. 



We have yet to introduce v in place of y : for this, observe that dy in the above element 

 supposes u constant. We have then dy = y v dv and the final result of the transformation is 



(x u y v - w vyJ xdudv - 



* See also Lagrange (Mem. Acad. 1773, p. 125). I have I preference to Lagrange, whose case has three, 

 cited Legendre, whose case has two independent variables, in 



