OF FLUIDS ON THE MOTION OF PENDULUMS. 



[27] 



and Q are functions of r and 8 only. An artifice, however, which has been extensively 

 employed by M. Cauchy will here be found of great use. Instead of introducing the circular 

 functions sinnt and cos nt, we may employ the exponentials e^i"', and f-V^Inf. Since 

 our equations are linear, and since each of these exponential functions reproduces itself at each 

 differentiation, it follows that if all the terms in any one of our equations be arranged in two 

 groups, containing as a factor e^^"' in one case, and e-^J-int in the other, the two groups 

 will be quite independent, and the equations will be satisfied by either group separately. 

 Hence it will be sufficient to introduce one of the exponential functions. We shall thus have 

 only half the number of terms to write down, and half the number of arbitrary constants to 

 determine that would have been necessary had we employed circular functions. When we have 

 arrived at our result, it will be sufficient to put each equation under the form U + \/~-\ V= 0, 

 and throw away the imaginary part, or else throw away the real part and omit \/ — l since 

 the system of quantities U, and the system of quantities V must separately satisfy the equa- 

 tions of the problem. Assuming then 



dt 



ce 



V-lnt 



y ■ e F > 



we have to determine P as a function of r and 6. 



15. The form of the equations of condition (SO) points out sin 2 (9 as a factor of P, and 



since the operation sin 6 — - — — - — - performed on the function sm s 6 reproduces the same 



ad sin ad 



function with a coefficient - 2, it will be possible to satisfy equations (27) and (28) on the sup- 

 position that sin 2 (3 is a factor of yi and y 2 *. Assume then 



yi = e VZ1 " f sin 2 6 % (r), y 2 = e V ~ lM < sin 2 0/, (r). 

 Putting for convenience 



n \/- 1 - n'm 2 , (32) 



and substituting in (27) and (28), we get 



f"(r) " ^/.W - 0, ....... . (33) 



f2"(r)--J. 2 (r)-m?Mr) = (34) 



r 



* When this operation is performed on the function 



dYi 

 sinB—, the function is reproduced with a coefficient 



- » (« + 1). Yi here denotes a Laplace's coefficient of the i ,h 

 order, which contains only one variable angle, namely 6. 

 Now <// may be expanded in a series of quantities of the 

 >dYi 



general form sin 6 



<I0 



For, since we are only concerned 



with the differential coefficients of ty with respect to r and 

 6, we have a right to suppose i^ to vanish at whatever point 

 of space we please. Let then \jr = when r = a and 6 = 0. 

 To find the value of \fi at a distance r from the origin, along 

 the axis of x positive, it will be sufficient to put 6 = 0, d8 = 

 in (26), and integrate from r = o to r, whence i^ = 0. To 



find the value of i^ at the same distance r along the axis of 

 * negative, it will be sufficient to leave r constant, and in- 

 tegrate d tj/ from 6 = to 6 = tt. Referring to (26), we see 

 that the integral vanishes, since the total flux across the 

 surface of the sphere whose radius is r must be equal to zero. 

 Hence \j/ vanishes when 6 = or = ir, and it appears from 

 (26) that when 6 is very small or very nearly equal to nr, \j/ 

 varies ultimately as sin* 6 for given values of r and t. Hence 

 \\r cosec 6, and therefore f\]r cosec 6 dS, is finite even when 

 sin 6 vanishes, and therefore /<//cosec6rf6 may be expanded 

 in a series of Laplace's coefficients, and therefore if> itself in 



dYi 

 a series of quantities of the form sin 6 — — - . It was somewhat 



06 



in this way that I first obtained the form of the function \fr. 



28—2 



