PROFESSOR STOKES, ON THE NUMERICAL CALCULATION, ETC. 167 



give no indication of the law of progress of the function, and do not even make known what 

 the function becomes when x = qt . These series present themselves, sometimes as develope- 

 ments of definite integrals to which we are led in the first instance in the solution of physical 

 problems, sometimes as the integrals of linear differential equations which do not admit of 

 integration in finite terms. Now the method which I have employed in the case of the 

 integral W appears to be of very general application to series of this class. I shall attempt 

 here to give some sort of idea of it, but it does not well admit of being described in general 

 terms, and it will be best understood from examples. 



Suppose then that we have got a series of this class, and let the series be denoted by y or 

 f(x), the variable according to ascending powers of which it proceeds being denoted by x. 

 It will generally be easy to eliminate the transcendental function f(x) between the equation 

 V = f(' t ') an d it s derivatives, and so form a linear differential equation in y, the coefficients in 

 which involve powers of x. This step is of course unnecessary if the differential equation is 

 what presented itself in the first instance, the series being only an integral of it. Now by 

 taking the terms of this differential equation in pairs, much as in Lagrange's method of ex- 

 panding implicit functions which is given by Lacroix*, we shall easily find what terms are of 

 most importance when x is large : but this step will be best understood from examples. In 

 this way we shall be led to assume for the integral a circular or exponential function multiplied 

 by a series according to descending powers of x, in which the coefficients and indices are both 

 arbitrary. The differential equation will determine the indices, and likewise the coefficients in 

 terms of the first, which remains arbitrary. We shall thus have the complete integral of the 

 differential equation, expressed in a form which admits of ready computation when x is large, 

 but containing a certain number of arbitrary constants, according to the order of the equation, 

 which have yet to be determined. 



For this purpose it appears to be generally requisite to put the infinite series under the 

 form of a definite integral, if the series be not itself the developement of such an integral which 

 presented itself in the first instance. We must now endeavour to determine by means of this 

 integral the leading term in f(x) for indefinitely large values of x, a process which will be 

 rendered more easy by our previous knowledge of the form of the term in question, which is 

 given by the integral of the differential equation. The arbitrary constants will then be deter- 

 mined by comparing the integral just mentioned with the leading term in f(x). 



There are two steps of the process in which the mode of proceeding must depend on the 

 particular example to which the method is applied. These are, first, the expression of the 

 ascending series by means of a definite integral, and secondly, the determination thereby of 

 the leading term in f{x) for indefinitely large values of m. Should either of these steps be 

 found impracticable, the method does not on that account fall to the ground. The arbitrary 

 constants may still be determined, though with more trouble and far less elegance, by calcu- 

 lating the numerical value of/ (x) for one or more values of x, according to the number of 

 arbitrary constants to be determined, from the ascending and descending series separately, and 

 equating the results. 



• Traite du Calcul, &c. Tom. I. p. 104. 



