386 J. H. ROHRS, ESQ. ON THE 



Professor Stokes was kind enough to work out for me the first two or three roots of the 

 equation tan I — J = ( — J ; which with his method I shall here transcribe. It is clear that 

 q will always be nearly expressed by (2i + l), i being a positive integer. 



.-. cot 9 = (2i + l) — - 9. 



v ' 2 



This can be solved readily by successive substitution, or by trial and error, the first two 

 roots give 



6 = 12°. 34', 

 9= 7°. 23'. 



The second of these values corresponds tov24.2cg' — and would give us y/z^AScg — , 



which only differs by 2 in the fourth figure of the root from the value we have already found. 



The reason why I have chosen vibrations in which only one type appears will be given 

 when we consider the action of small periodic forces on the chains ; it will then be seen that 



in bridges of wide span terms containing the types \/24. 2cg — are very likely to be those 



which express the disturbance occasioned by soldiers marching in time along the platform ; 

 and, consequently, that I could scarcely have selected a better example to illustrate the subject, 

 generally, and the effect of the small terms I shall hereafter include — in particular — than 

 the one we are now discussing. 



To determine the tension we have 



6? dV I s*\ <PV s . 



^^ = ^l 1 + 2v)d7 + M, ; from(1) ' 



. . „ dy . , s j s , \ 



writing for — its value -, — . exactly ; 



& ds c'V^T? / 



s 2s d 2 



,\ w - = r -77 (F- M ti ) and when s = a, 



c & dt 



and - . — — - M = - 23 . 2cg — . 85A cos s/cg . 24 . 2 — - . 

 c 2 7r at 4«* 2a 



The term in v or — — ■ corresponding to A 3 is 

 ds 2 



— • h cos \/cs . 24 . 2 — t, let therefore — h = h', 

 2a Y s 2a 2a 



and we shall find the tension - 122 . h'g cos (\/cg . 24 . 2 — t) nearly. 



