390 Sir "William Thomson on the General Integration 

 and theiij eliminating a, h, H from (5), (7), (8), 



?{ 



d 



< 



sin0 



du 2n 



la + s — 

 ad a 



cosOu) 



sin 



a 2 -4n 2 cos?0 

 (2n cos du 



syy- 



~0 



+ 



su 



sm' 



o- 2 _4?z 2 cos 2 <9 



+ m=-E. (9) 



This is Laplace's differential equation of the tides [Mecanique 

 Celeste, Liv. IV. No. 3, equation (4); or Airy, "Tides and 

 Waves/' Encyclopedia Metropolitana, art. (95)]. It is a linear 

 differential equation of the second order, the complete integration 

 of which gives u, and thence, by (8), a and b, in terms of 0, with 

 two arbitrary constants to be determined so as to fulfil proper 

 terminal conditions (§§11-17, below) . It is essentially in the form 

 in which Airy gave it, being that in which it comes direct from 

 the formulae preceding it in the investigation. It originally ap- 

 peared in the Mecanique Celeste, masked somewhat by the addi- 

 tion and subtraction of a certain term which gives it a different 

 form, not seeming at first sight better or simpler ; but this as it 

 were capricious modification suggests the following very sub- 

 stantial simplification, 

 4. Put 



and 



then we have 



(sin0) o- ££==<£ 



2ns 



(sin0)VE = <I> tJ 

 d<j> 



(10) 



a=~ (sm#) <r t *-- 



r K J c7 2 — 4?z 2 cos 2 0' 



2n 



*=-f(Bin0)- 





cos 6 



d<f> 



I (11) 



sin0d0 



2 -4n 2 cos* 6 



s<f> 



sin 2 



}■■ 



If (with Laplace) we put cos 0=fJb, and for brevity 



irr -\ 

 — = m 



9 



and 



2n~f'-< 

 these equations (11) become 



(12) 



