164 Mr. Ivory on such Functions as can be 



supposing that all the quantities under the double sign of in- 

 tegration are expressed in terms of the two variables 7/ and z, 

 and that « is made equal to 1, after the integrations. This 

 transformation is correct; and the introducing of the new- 

 integral is important, as it leads to detecting the fault of the 

 investigation. M. Poisson assumes that the differentials of 

 the new integral X" are all infinitely small, so long as ? is in- 

 finitely small: in which case, X" being itself infinitely small, 

 it may be rejected, and the same conclusion will be obtained 

 as in the first investigation. Now were the denominators in 

 the successive differentials of X" always finite quantities, the 

 assumption of M. Poisson would be allowable; but as both 

 the numerators and denominators begin to vary from zero, it is 

 not impossible that, while the first increase to a finite magnitude, 

 and the others to some small quantity B, the quotients may 

 pass through every gradation of quantity; their values may 

 be infinitely small, or finite, or infinitely great; This point 

 must therefore be examined before any just conclusion can be 

 drawn. 



Let u =/{6, \J/), u' =/(fl', ^') ; then ^ = u'—u : put also 

 g = 1 — a; then 



1 - «2 = 2 g - g- 



1— 2aj5 + «^ = 2(1— p) — 2g(l— p)-[-g^ 



These values being substituted, the resulting expression of 

 X" will be, 



, _ fyf' {2 g^g'). u'-u.sm 6' d^'d^'^ 

 Jo Jo (2 (1-p) — 2g(l-^)+gO^ 



Now 1 —p is a small quantity depending upon the values 

 assigned to 1/ and z; and g is a small quantity quite inde- 

 pendent of any other ; we may therefore suppose that g is 

 equal to 1 — p, or less than it, and even infinitely less than it. 

 Now, c being any positive number less than |, if we reject 

 quantities of the second order in the last formula, the result 

 may be thus written : 



X"= r^r^^-^ X "'"''■_ xsmS'dQ'd'^iy', 



Jo Jo (2{l-p)y {2(l-p))i " 



which is obviously the limit to which the expression of X" 

 continually tends as values decreasing indefinitely are assigned 

 toj/, 2, g. Since g may be considered infinitely less than 



1 —p, the ultimate value of the factor -^ — is always 



(2(1-;,))^ 



