of a Solid Body in a Viscous Fluid. 631 



The method of formation of 6{x) is clear from the equa- 

 tions for a finite number of terms ; multiply the left-hand 

 side of (12) by yjr(x) and a factor to make the value unity 

 for x zero. From (11) and (12), we have 





(13) 



Writing the roots of (12) as ot= — vX 2 \h 2 and collecting 

 the results from (6), (12), and (13), the first step in the 

 solution of (3) gives 



dhj 2 fyptft f ^\*e-*>*«- T )l>* nA s 



-J =fy- -^-J/WS^qr^irr)^ • (14) 



where the summation extends over the positive roots of 



X tan X— 2ph/a = k (15) 



Following the method of reduction for this type of equa- 

 tion*, integrate (14) with respect to t from to 0, using 

 Dirichlet's formula to transform the order of integration 

 of the last term. Since the initial value of dyjdt is zero, this 

 leads to 



Integrate (16), in the same manner, with respect to 

 from to T ; finally, for convenience, replace T by t and 

 t by r, respectively, in the result. Then we obtain 



^) + J7«L^ w (17) 



The solution of (17) can be completed by means of (6), 

 (8), and (11). The new exponents are given by 



2? + V?S 1 4.1-0 (18) 



x ^ <rv 2 *\*{\* + k(l + k)\(x + v\*lh*)* [ } 



Resolving the summation into one of simple partial frac- 

 tions and using the properties of the roots of (15), this 

 equation can be reduced to 



x 2 + — coth — r -f/)- = 0. . . . (19) 



* Volterra, 'Lecons sur les Equations Integrates,' p. 140. 



