TRANSACTIONS OF SECTION A. 475 



(9), after having multiplied theni respectively by /3.,, /3 3 , ..., /3„. In the same manner 

 the second is obtained by means of the multipliers y._„ y 3 , ..,, y n and so on until the 

 last, which is obtained by using k. 2 , k v ..., k k as multipliers. The equations, thus 

 formed, are in virtue of the notations (1) : 



' (fl^D, + (&/3),D 2 + (c^D 3 + ... + (A^D,, = o, 

 («y) 1 D I + (by),!), + (cy)^ + ... + (/cy)^ = 0, 



(10) 



. (rt^jDj + (MiDo + (ck) x D s + ... + (Jut) fin = o. 

 From the equations (8) and (10) we get by elimination of D 2 , D 3 , ..., D, 



(bftv 0/3)i> •••> (*0)i 

 ( & y)i. (cy)u ■-, (h)i 



V.D 1 = D. 



Wd ( Ck )i> ■■■> (**)i 



The last determinant in the second member of this equation is = Aj, in virtue 

 of the well-known fundamental theorem about the change of rows into columns. 

 The last equation, therefore, gives — 



whence, because by supposition the identity (5) holds, 

 we get 



V = D. Aj. 



Substituting this expression for v in (6), there results 



A = D . A, q.e.d. 



18. Un the Motion of Two Cylinders in a Fluid. By W. M. Hicks, M. A. 



The motion of two infinite parallel cylinders in an infinite fluid, and the motion 

 of one cylinder inside another full of fluid, were considered. In the second case, 

 when the inside cylinder (rad = b) moved as a pendulum about the axis of the 

 bounding one (rad = a), it was shown that the time of vibration was changed in the 

 ratio 



(t7h^{ M+3M >832~ S }> 



where M, M x are the masses of unit of length of the cylinder, and of the fluid 



displaced thereby ; x = and L depends on the distance between the centres. 



a — b 



In the particular case where the inside cylinder touches the outside throughout 



the motion it was shown that 



L = l + 2.t- 2 -^logr(l + .r) 



tables of which function are given by Legendre and De Morgan. 



The more general problem was then discussed, and formulas obtained for the 

 motion of an infinite fluid in which two cylinders rest, and in which a source of 

 fluid exists. Thence the velocity potential for any motion of the cylinders was 

 deduced, and the kinetic energy of the fluid. It was shown that (w, vf ; v, v', 

 being the velocities along and perpendicular to the line joining the centres of the 

 cylinders), the kinetic energy was — 



2T = Pi* 2 + (tto 2 + P) v 2 + PV 2 + (wb 2 + P>' 2 + 2L(w' - im'), 

 where 



~P = 2ira 2 y(6,0 l ), 



P' = 2,r& 2 y (]•£,), 



L = 2 7 r«&y 1 (^)» 



