OF FLUIDS ON THE MOTION OF PENDULUMS. [19] 



the motion of the fluid has been determined in terms of the quantities by which the motion of 

 the pendulum is expressed. 



5. Before proceeding to the solution of the equations (2) and (3) in particular cases, it 

 will be well to examine the general laws which follow merely from the dimensions of the several 

 terms which appear in the equations. 



Consider any number of similar systems, composed of similar solids oscillating in a 

 similar manner in different fluids or in the same fluid. Let a, a', a"... be homologous lines in 

 the different systems; T, T', T" ... corresponding times, such for example as the times of 

 oscillation from rest to rest. Let x, y, % be measured from similarly situated origins, and in 

 corresponding directions, and t from corresponding epochs, such for example as the com- 

 mencements of oscillations when the systems are beginning to move from a given side of the 

 mean position. 



The form of equations (2), (3) shews that the equations being satisfied for one system will 

 be satisfied for all the systems provided 



fill pUX 



tt cc l> cc 10, X cc y o: X, and p cc ee . 



X t 



The variations x cc y « x merely signify that we must compare similarly situated points in 

 inferring from the circumstance that (2), (3) are satisfied for one system that they will be satis- 

 fied for all the systems. If c, c, c" ... be the maximum excursions of similarly situated points 

 of the fluids 



c 



tta— I Sec H, t cc T, 



and the sole condition to be satisfied, in addition to that of geometrical similarity, in order 

 that the systems should be dynamically similar, becomes 



— cc - or cc fi (6) 



r P 



This condition being satisfied, similar motions will take place in the different systems, and we 



shall have 



pac 

 P~^fi (7) 



It follows from the equations (4), (5), and the other equations which might be written 

 down from symmetry, that the pressures such as JP U T s vary in the same manner as p, whence 

 it appears from (7) that the resultant or resultants of the pressures of the fluids on the solids, 

 acting along similarly situated lines, which vary as pa 2 , vary as pa 3 and cT~ 2 conjointly. 

 In other words, these resultants in two similar systems are to one another in a ratio com- 

 pounded of the ratio of the masses of fluid displaced, and of the ratio of the maximum 

 accelerating effective forces belonging to similarly situated points in the solids. 



6. In order that two systems should be similar in which the fluids are confined by 

 envelopes that are sufficiently narrow to influence the motion of the fluids, it is necessary that 



27—2 



