86 THE ROYAL SOCIETY OF CANADA 



The first values of u to be considered as possible singularities of 

 X (u) are those values Uj of u at which x = a^, 0,2, ag, a^, or 00, 

 for the expansions of \/R(x) in terms of x — a,, x — a,^, etc. do not 

 conform to the conditions imposed on f (x, u). We have seen, already, 

 in another connection that the proper way to treat the neighboui hoods 

 of these points is to use the transformation x - aj = t", or the trans- 

 formation X = 1 /t. When this is done the resulting differential equation 

 in dt/dx conforms to the requirements of Cauchj^'s theorem. Hence 

 t - o and therefore x - aj or 1/x, are expressible in the neighbourhood 

 of such points by power series in u - u j. Observe that in the latter case 

 (that of 1 /x) the expression for x is a power series preceded by a polar part. 



If these values Uj were the only ones to be taken into account, it 

 would be legitimate to conclude, without further discussion, that the 

 whole finite u-plane can be covered by continuations from the primary 

 power-series, except those points u^ for which u is infinite. Thus x (u) 

 would have been proved to have the character (which it actually has) 

 of a function that is meromorphic over the whole of the finite u-plane. 

 This conclusion was drawn by Briot and Bouquet. 



In speaking of the values u 1 that lead to singularities of x (u) we 

 have assumed that the analytic function x (u), derived by the process 

 of continuation, must take a determinate value at each point of the 

 plane; briefly, to every u an x. The only u's that required special 

 examination were those associated with values x (u) which were = ai, 

 &2, aj, a^, 00. But this overlooks the possibility that a singular point Uj 

 might have no determinate value of x corresponding to it. 



Singularities of this kind do not present themselves in our case of 

 the elliptic differential equation, but as soon as we pass to differential 

 equations of the second order, the ignoring of these singularities might 

 lead and has led to erroneous results. Painlevc's equation— 



d^x /dx\2 / 0x2-^-2 1 



:)( 



du- \du/ V4x^ - goX - g, V 4x3 - goX - gg / 



of which the solution is x = p (a -1- log (u -f- b) ) , where a, b are arbi- 

 trary constants, illustrates the points involved. Let Uj be any finite 

 value and xj, xl, any two values, finite or infinite; if a solution Xj (u) 

 tends to the limit x\, and Xj (u) to the limit Xg, when u tends to u 1 along 

 a certain path L, then the function is" certainly holomorphic or mero- 

 morphic in the neighbourhood of u ,. If now we make the same inference 

 that Briot and Bouquet made in the simpler case, we should conclude 

 that p (a+log (u+b)) is one-valued, whereas for r/c/!craZ periods of p it 

 is infinitely many-valued. The weak point in the reasoning lies in the 

 assumption that some value of x must correspond to u ; for where u = 

 - b, the integral is indeterminate. 



