,1 



246 Prof. W. M. Thornton on the Cause of Lowered 



due to viscosity, if taken as usual to be proportional to rate 

 of displacement, leads to results not in keeping with the- 

 experimental facts. 



Writing equation (1) with a viscosity term 



doc 

 a -j- + bx— ex? 2 = F sin pt. 



(9) 



This equation was submitted to Professor A. P. Forsyth T 

 who very kindly gave the following note upon it : — 



"The equation is not integrable in 'finite terms/ It 

 belongs to the Riccati type, and its integral can best be 

 expressed by means of series. Take 



a 1 du b 

 c u at 2c 

 the equation becomes 



dt 2 

 Let pt = -ir + 26, and 





4a 2 

 write sin 2 6 = z 



> 



0. 



. . . (10) 



'. ■ ■ (11) 



the equation 



becomes, first 



d0* 



and then 



z- N dhi 



\a?p 2 



+ '~^ cos ^-^v) 



o, 



(12) 



Of this equation there are two distinct integrals proceeding 

 in ascending powers of z, which is positive and not greater 

 than 1. One of these is of the form 



and the other 



l + h 1 z + h 2 z 2 + ... =T 1? 



and the most general value is 



u = AT, + BT 2 , 



2? 



• • (11) 

 . . (15) 



■ • (16) 



where A and B are arbitrary constants. Thus the value of 

 x would be 



b a dt 



dT 1 + B dT, 

 dt 



c AT 1 + BT 2 ' ' * 



which really contains only one arbitrary constant . 



(17) 



