OF ARBITRARY CONSTANTS, &c. 117 



traced, since F^ (0) = Fi(9 + Stt). If now we conceive the curve marked with the proper 

 values of the constants G, D, it will serve to represent the complete integral of equation (18). 



In marking the curve we may either assume the amplitude 6 of a? to lie in the interval 

 to 27r, and determine the values of C, D accordingly, or else we may retain the same value of 

 C or D throughout as great a range as possible of the curve, and for that purpose permit 9 to 

 go beyond the above limits. The latter course will be found the more convenient. 



16. We must now ascertain in what, cases it is possible for the constant G or D Xo 

 alter discontinuously as 9 alters continuously. The tests already given will enable us to 

 decide. 



The general term of either series in (20), taken without regard to sign, is 



1.5... (6i-5)(6t-l) 

 1 .2... i(144.r*)* ' 



and the modulus of this term, expressed by means of the function F, is 



r (e + j) r (i + 1) 

 r(^)r(f)r(i + i)(vt)'' 



which when i is very large becomes by the transformations employed in Art. 7, very nearly, 



\/y(;)Vr(i)r(i)(4/,i)*. 



Denoting this expression by /Ujj and putting for r(^) T{^) its value Trcosec- or 27r, 



we have 



"^-^^""'^''Wef'' ^''^ 



whence for very large values of i 



—-A (24) 



IXi 4p« 



For large values of p the moduli of several consecutive terms are nearly equal at the part 

 of the series where the modulus is a minimum, and for the minimum modulus n we have very 

 nearly from (24), (23) 



i = 4pi, fi = (27ri)-^e-* = (2ni)~ie~*''*. 



If the exponential in the expression for j^i be multiplied by the modulus of the exponential 

 in the superior term, the result will be 



g-(4T2c08| 9)p* 



the sign — or + being taken according as cos § ^ is positive or negative. Hence even if the 

 terms of the divergent series were all positive, the superior term would be defined by means of 

 its series within a quantity incomparably smaller, when p is indefinitely increased, than the 

 inferior term, except only when =tcos50= 1, and in this case too and this alone are the 

 terms of the divergent series in the superior term regularly positive. In no other case then 



