SIMILAR TO THOSE OF HYDRODYNAMICS. 



387 



Now we have found that 



J?A n _ du du du, 



~W ~ A * ^ + A * d^ + •••• + An ^ 



Using this equation, and also equation (103) and we are enabled to write at once 



DA n _ du x 

 dt ~ ' d d^ 



dt ~~ J dJ x 



(112) 



DA ln _ du„ 

 dt ~ J d6 1 



Multiplying these by « l5 i< 2 , . . . . u n respectively and adding the sum to (111) we 

 obtain 



•I 



dW 

 d0 x 



there are of course n 

 written down. Since 



1 other equations of this form which can be immediately 



DJ 



~dt 



= 



we have, if we write P 



DW 



dt 



integrate and solve the resulting equations for 



111, U i 5 



. u„ the same expressions as those marked (106). 



Note. — I desire here to call attention to the paper published by Professor Rowland in the American 

 Journal of Mathematics, Vol. III. No. 3, in which a full account of the so-called higher orders of motion 

 is contained. (I may also mention that I am informed by Professor Rowland that he gave a brief account 

 of these motions at the meeting of the Scientific Association of the Johns Hojikins University on 

 October 6th). 



The existence of these motions was discovered at about the same time independently by Professor 

 Rowland and myself. He was led to consider them while treating of the equations of electromagnetism, 

 and their connection with fluid motion. I was led to similar results by the study of certain possible cases 

 of steady motion in viscous fluids, an account of which will be found in the Journal of the Franklin 

 Institute for October, 1880, and also in the Philosophical Magazine for November, 1880. 



