630 TRANSACTIONS OF SECTION A. 



As I believe the method adopted iu this to constitute the simplest and most 

 satisfactory solution of the problem, I would endeavour in the present paper to 

 present it in a modified form, which may perhaps more clearly bring out its 

 scope and spirit. The difficulty to be overcome arises, as is well known, from the 

 fact that though, assuming a velocity potential, the velocities of any point .r, y, z 

 of the fluid are determined in terms of the q\, q'n-.. representing the motions of 

 tlie solids, or assuming a displacement potential, the displacements of :i', y, z 

 are determined in terms of dqi, dq.^ ... it is not possible, as is required for the 

 applicability of the Lagrangian equations, to express the coordinates a; y, z them- 

 selves in terms of y,, y.j . . . 



Now, although this cannot be done, we may, starting from a given configu- 

 ration of the solids supposed at rest, and connecting y,, q.-, as functions of a single 

 variable, express iu terms of this variable the position of a given point ,r, y, z. 

 AVe may, in fact, conceive the following process ; Sudden connected velocities 

 q^,q.i . . . are imparted to the solids, determining corresponding velocities of .r',!/,^. 

 After an indefinitely small time the solids are stopped, and therefore .r, «/, s is also 

 brought to rest. The process is then repeated with new connected velocities 

 q\ . . , producing a new velocity of x, y, z again reduced to rest. In this way, 

 by what we may term a continuous system of jerks, ,r, y, z is continuously dis- 

 placed by 



dx , . d,v , 



— - f/y, . dq., . . . 



we must have regard to the generalised coordinates q, not only as they occur in 

 q,^ but as they arise from the variations of x, y, z, i.e., we follow the particle. 

 Consider now the virtual moment which in general gives rise to the Lagrangian 

 equations, viz. : — 



V^ /d'^x , <£-« , d-z 



/d\v , d-y , d-z \ , 



For the solids r, y, z being now functions of q^, q.,, this yields the Lagrangian equa- 

 tion. For the third we throw the virtual moment into the form 2 Qmdqm, where Q,„ 



^ \ dt^ dq^ dt' dq^ dt' dqr, J 



Now 



dq„ de dq^ df dq„ 



d^x dx_ _ d^ dx dx _ dx d dx 

 df^ dq„ ~ dt ' di dq^ dt dt dq„ 



with analogues y, z. 



The former term, since -— ^ = — „— —, gives rise to ^ — 



dqm dq m dt dt dq^ 



