OF ARBITRARY CONSTANTS, &c. 123 



Hence the equation (^33) may be written, according to the notation employed in Art. l6, 

 as follows : 



M = (0 to Stt) Cw-ie'Ul + ...) + (-TT to +7r)Z)a?-^e'(l + + ...) (34) 



2. 4a? 2.4.J? 



22. It remains to connect A, B with C, D. For this purpose we shall require the third 

 form of the integral of (31), namely 



u 



= f^{E+ F\og{a>sm''w)\(g'"'"'+e-'''°"')dw (35) 



(36) 



As to the value of logx to be taken, it will suffice for the present to assume that whatever 

 value is employed in (32), the same shall be employed also in (35). 



To connect A, B with E, F, it will be sufficient to compare (32) and (35), expanding the 

 exponentials, and rejecting all powers of x. We have 



A + B logx = 2 j [E + F log (a> sin'^w) \ dw 



= 7r(£+i^log.r) +27rlog(i).F; 

 whence 



^ = TT^ - 27r log 2 . F,l 



B = 7rF. J 



To connect C, D with E, F, we must seek the ultimate value of u when p is infinitely 

 increased. It will be convenient to assume in succession 6=0 and 9 = tr. We have ulti- 

 mately from (34) 



u = Dp-ie^ when 6 = 0; u =- \/ - 1 Cp'hi' when 6 = ir (37) 



It will be necessary now to specify what value of log a; we suppose taken in {S5). Let it be 

 log p + \/ — l6, 6 being supposed reduced within the limits and 2ir by adding or subtracting 

 if need be 2i7r, where i is an integer. 



The limiting value of u for 6 = from (35) may be found as in Art. 29 of my paper on 

 Pendulums, above referred to. In fact, the reasoning of that Article will apply if the 

 imaginary quantity there denoted by m be replaced by unity. The constants 



C, D, C, D', C", D", 

 of the former paper correspond to 



A, B, C, D, E, F, 

 of the present. Hence we have for the ultimate value of u for = 



M=[^]V{i;+(7r-ir'(J)+log2)/'| (38) 



For = TT, i^5) becomes 



IT 



u= f''{E+ TrFy/- 1 + /'log (p sm^ w)l (e-'"=<''"+ e^~'") dw ; 



Jo 



and to find the ultimate value of u we have merely to write E + irF y/— 1 for E in the above, 

 which gives ultimately for = tt 



^ = (^y^l^ + '^^y/^^ + {-^'^^'(D + ^^s^l ^ (39) 



16—2 



