Solid Bodies through a Liquid. 337 



linear equations, of which (17) is an abbreviated expression, and 

 so have Q expressed as a quadratic function of yfr, <j>, 6, . . . , 

 with its coefficients functions of \fr, <j>, 6, &c. ; and if we denote 



by -j-r) -j—-3 &c. variations of Q depending on variations of these 



coefficients, and by — .-> —. , &c. variations of Q depending on 



d\jr dcf> 

 variations of yjr, $, &c, we have [compare Thomson and Tait, 

 § 329 (13) and (15)] 





fo= 



dQ, 





dQ 





d$ 



%' 



~ df y ' 



and 













m 





dQ 



mQ 





df 





dtf 



d(f) 



and the 



equations of motion become 



dQ 

 dp * * • 



(21) 





dyjr 



..=<!>- 



d_dQ 

 dtdd 



dQ 

 dd 



{ft Wf -{ft £}£+... ,===©- 



d<t> 



..(22) 



The first members here are of Lagrange's form, with the remark- 

 able addition of the terms involving the velocities simply (in 

 multiplication with the cyclic constants) depending on the cyclic 

 fluid motion. The last terms of the second members contain 



traces of their Hamiltonian origin in the symbols -j-j-j — , . . . 



8. As a first application of these equations, let ^ = 0, <£ = 0, 

 0=0, . . . . This makes f =0, ?/ =0, . . . , and therefore also 

 Q = ; and the equations of motion (16) (now equations of equi- 

 librium of the solids under the influence of applied forces'^, <3>, 

 &c. balancing the fluid pressure due to the polycyclic motion 



Ky (C , 



become 



-.s r-5<** 



(.23) 



a result which a direct application of the principle of energy 

 renders obvious (the augmentation of the whole energy produced 

 Phil. Mag. S. 4. Vol. 45. No. 301, May 1873. Z 



