OF ARBITRARY CONSTANTS, &c. 109 



tain the factor p~^'^, yet as the mutual destruction of positive and negative parts may take 

 place quite differently in the two integrals jie^'da and fAfC^'da, we can conclude nothing as 

 to their relative importance. 



6. Now cos 20 will continually increase or decrease from one limit to the other, or else 

 will become a maximum, according as the two limits 0^ and lie in the same interval to tt 

 or TT to 27r, or else lie one in one of the two intervals and the other in the other. Hence we may 

 employ the expression (4), with an invariable value of C yet to be determined, so long as 

 < < TT, and we may employ the expression obtained by writing C' for C so long as 

 IT <6 < Stt, but we must not pass from one interval to the other, retaining the same expression. 

 Now we have seen (Art. 3) that the constant changes sign when is increased by tt, and there- 

 fore C' = - C And since u is unchanged when is increased by any multiple of 27r, we readily 

 see that in order to make the expression (4) generally applicable, it will be sufficient to change 

 the sign of the constant whenever d passes through zero or a multiple of ir. 



7. We may arrive at the same conclusion in another way, which will be of more general 

 or at least easier application, as not involving the integration of the differential equation. 



The modulus of the general term (Art. 4) of the series (4), expressed by means of the 

 function F, is 



r(i)p^*+'- 



Suppose i very large. Employing the formula 



r (a? + l) = v 27r« (- I » nearly, when a? is large, 

 observing that r(^) = tt^, and calling the modulus yUj, we find 



which, since (i + cf = i'e", nearly, becomes 



Me = 2*i«e->-^'-i (7) 



We easily get, either from this expression or from the general term, 



,'^=^, nearly, (8) 



Mi P 



Hence when p is large the ratio of consecutive moduli becomes very nearly equal to unity 

 for a great number of terms together, about where the modulus is a minimum. To find 

 approximately the minimum modulus n, we must put i = p^ in (7), which gives 



M-gV'e-"' (9) 



If we knew precisely at what term it would be best to stop, the expression for n would 

 be a measure of the uncertainty to which we were liable in using the series (4) directly, that 

 is, without any transformation. For although it is clear that we must stop somewhere about 

 the term with a minimum modulus, in order that the differential equation (5) which our 

 function really satisfies may be as good an approximation as can be had to the true differential 



