OF FLUIDS ON THE MOTION OF PENDULUMS. [33] 



to determine the four arbitrary constants which appear in (38) by the four equations (35) and 

 (36). We get, in the first place, 



- + Ba* + Ce- ma ( 1 +— } + De na f 1 - — ) ^la'c, . ... (54) 



a \ ma) \ ma) " v ' 



--+2Ba*-Ce- ma (ma + l +— ) + De ma (ma-l+~) = a 2 c, . . (55) 

 a \ ma) \ ma) 



^ + ^ + C 6 -(i + ^) + ^(l-^)=o, (56) 



-^ + 2Bb*-C6- mb (mb + l + -^~) +De mb (mb-1 + ~) = o. . . (57) 

 b \ mb) \ mb) 



Putting a % cK for af((a) +2/ 2 (a), which is the quantity that we want to find, we get 

 from (38) and (54) , 



* ml ~c < 58 ) 



Eliminating in succession B from (54) and (55), from (56) and (57), and from (54) and (56), 

 we shall obtain for the determination of A, C, D three equations which remain unchanged 

 when a and b are interchanged, and the signs of A, C, and D changed. Hence - A, - C, -D 

 are the same functions of b and a that A, C, D are of a and b. It will also assist in the 

 further elimination to observe that C and D are interchanged when the sign of m is changed. 

 The result of the elimination is 



K m . _ _£6 gfo b ) - v(b, a) 



<Zm*a*' 12mab + £(a,b) + £(b,a)' ^ 



the functions £, r\ being defined by the equations 



n (a, b) = (»reV + 3ma + 3) (m s 6 2 - 3mb + 3) 6 m (*-°), 1 



£(a, 6)= \b(m*V-3mb+3)-a(m ! a i + 3ma + 3)\ e m ( b - a ). J (60) 



It turns out that Kiss, complicated function of m and ab~ l , and the algebraical expressions 

 for the quantities which answer to k and k' in Art. 20 would be more complicated still, because 

 n(l +\/ - 1) would have to be substituted for m in (60) and (59), and then K reduced to the 

 form -fc + v-lA;'. To obtain numerical results from these formulae, it would be best to 

 substitute the numerical values of a, b, and v in (60) and (59), and perform the reduction of 

 K in figures. 



22. If the distance of the envelope from the surface of the sphere be at all considerable, 

 the exponential e v ^ b ~ a \ which arises from 6 ra ( J - a ), will have so large a numerical value that 

 we may neglect the terms in the numerator and denominator of the fraction in the expression 

 for K which contain e ~ v ^ b ~ a \ as well as the term in the denominator which is free from expo- 

 nentials, in comparison with the terms which contain e"( b ~ a \ Thus, if b - a be two inches, 

 t one second, and y//x = .116, we have e v ( b - a ) = 2424000000, nearly ; and if b - a be only an 

 Vol. IX. Paet II. 29 



