OF ARBITRARY CONSTANTS, &c. 115 



13. Let p, 6 be the polar co-ordinates of a point in a plane, O the origin, C a circle 

 described round O with radius unity, S the point determined by a; = - 1, that is, hy p= 1, 

 = IT. To each value of x corresponds a point in the plane ; and the restriction laid down 

 as to the moduli of a> confines our attention to points within the circle, to each of which 

 corresponds a determinate value of u. If Pg be any point in the plane, either within the 

 circle or not, and a moveable point P start from Pg, and after making any circuit, without 

 passing through S, return to P, again, the functipn (1 + my will regain its primitive value m^,, 

 or else become equal to - u^, according as the circuit excludes or includes the point S, 

 which for the present purpose may be called a singular point. Suppose that we wished to 

 tabulate u, using when possible the divergent series (l6) in place of the convergent series (15). 

 For a given value of 6, in commencing with small values of p we should have to begin with 

 the series (15), and when p became large enough we might have recourse to (l6). Let OP be 

 the smallest value of p for which the series (l6) may be employed ; for which, suppose, it will 

 give u correctly to a certain number of decimal places. The length OP will depend upon Q, 

 and the locus of P will be some curve, symmetrical with respect to the diameter through S. 

 As 9 increases the curve will gradually approach the circle C, which it will run into at the 

 point 8. For points lying between the curve and the circle we may employ the series (l6), 

 but we cannot, keeping within this space, make 9 pass through the value tt. The series 

 (16), (17) are convergent, and their sums vary continuously with x, when p> \; and if \ve 

 employed the same series (l6) for the calculation of u for values of w having amplitudes 

 TT - /3, IT + fi, corresponding to points P, P', we should get for the value of m at P' that 

 into which the value of m at P passes continuously when we travel from P to P outside the 

 point S, which as we have seen is minus the true value, the latter being defined to be that 

 into which the value of u at P passes continuously when we travel from P to P' inside 

 the point S. 



In the case of the simple function at present under consideration, it would be an arbitrary 

 restriction to confine our attention to values of x having moduli less than unity, nor 

 would there be any advantage in using the divergent series (l6) rather than the convergent 

 series (15). But in the example first considered we have to deal with a function which 

 has a perfectly determinate and unique value for all values of the variable a, and there is 

 the greatest possible advantage in employing the descending series for large values of p, 

 though it is ultimately divergent. In the case of this function there are (to use the same 

 geometrical illustration as before) as it were two singular points at infinity, corresponding 

 respectively to = and 9 = n. 



14. The principles which are to guide us having been now laid down, there will be no 

 difficulty in applying them to other cases, in which their real utility will be perceived. I will 

 now take Mr Airy's integral, or rather the differential equation to which it leads, the treatment 

 of which will exemplify the subject still better. This equation, which is No. 11 of my paper 

 " On the Numerical Calculation, &c,," becomes on writing u for U, — 3^ for n 



— -9a7M = (18) 



15—2 



