^{page 138 note *} The same is true if a law is given by means of which one of the limits is denned for each value of y; e.g. the maximum limit. In this case a number of quantities can be identified as limits of limits of f(x, y), or repeated limits; e.g. the upper upper limit, the upper lower limit, etc. The whole set of such limits, which may theoretically be denoted by can only be regarded as perfectly denned when all possible laws which can be used are in some manner specified.

^{page 139 note *} These may be regarded as falling under the heading of simple limits, since they are , where F(n)f(x_n, y_n).

^{page 139 note †} It is unnecessary here to enter into the modifications necessary when the point (a, b) is at infinity.

^{page 142 note *} A proof of this is given in § 7 below.

^{page 142 note †} That is, not on x=a nor on y=b.

^{page 143 note *} Thinking of the representation in two dimensions, right and left with respect to y is, of course, “up and down.”

^{page 144 note *} Stolz, , Grundzüge der Differenzial- und Integralrechnuny, 1893, p. 147Google Scholar. Hobson, , “Partial Differential Coefficients and Repeated Limits,” Proc. L.M.S., 1906, series ii., vol. v. p. 234Google Scholar. See also Functions of a Real Variable, p. 318 and the errata. It should be noticed that the account in the book is really earlier than that in the paper.

^{page 144 note †} See footnote on preceding page.

^{page 145 note *} It should be noticed that the difference between Hobson's conditions, loc. cit., and those given by Schwarz consists in the omission of assumption (3), and that this is only possible in the light of the extended definitions suggested by Hobson. If such a definition be adopted, there is, it would appear, no reason for making the assumption (2) either.

^{page 148 note *} Or any isolated set of axial crosses; that is, any set of axial crosses such that, taking any point P internal to the region considered, we can find a region containing P as internal point and intersected by at most one of the axial crosses.

^{page 150 note *} The extension of this result to the case when f(x, y) is a quasi-finite function, which, in the case of one variable, was given in the paper quoted on p. 144, line 2, is not elaborated here, the attention being concentrated in this paper, in the first instance, on finite functions.

^{page 150 note †} That is, omitting the axial cross x = a and y = b.

^{page 151 note *} Closed neighbourhood.

^{page 151 note †} Open neighbourhood, omitting values at the point itself.

^{page 153 note *} Loc. cit.

^{page 153 note †} Line 13, loc. cit.

^{page 156 note *} See a paper by the author “On Non-differentiable Functions,” Mess, of Math., September 1908Google Scholar.