.354 Prof. Challis on the Motion of a small Sphere 



\ the breadth of an undulation : so that — x r- 1 * or -, is a small 



r x \ \ 



quantity of the second order. Hence, since qr may on this sup- 

 position be always taken to be very small, it is allowable to expand 

 the above sine in terms proceeding according to the powers of this 

 quantity. We shall thus have, to terms of the second order, 



V = a' o 1 = m' sin q (a't + c Q )-\-m' cos q (a't + c Q ) qr cos 



— sin q(a!t + c ) q^r 2 cos 2 0. 



The sole conditions that the integration of the equation (97) is 

 required to fulfil are, (1) that these approximate equations be 

 satisfied wherever r is very large compared to c and very small 

 compared to X ; (2) that U = where r=c, — that is, at the sur- 

 face of the sphere. Again, since the equation (77) is verified by 

 supposing P to be either c/> 1 sin 0, or <£ 2 sin cos 0, or the sum 

 of these two quantities, let us first suppose V = cj) l sin 0. Then, 

 as is shown in Part II., the following equations may be obtained, 



* =*M-(^-^')co S ; 

 W /7,-F. f-¥\ . , 



a' \ r 3 r z r / 



fififx being put respectively for / [r — a't), ~> and §fdr, and 



F, F, F t for F(r + aH), -p and JFdr. Since from the condi- 

 tions of the problem no part of <r can be a function of r without 

 0, the arbitrary quantity *ty (r, t) cannot contain r, but must be 



a function of t only. To determine this function, let 0— — . 



A 



Then, for all the corresponding values of r, a = '\jr(t) — a l suppose. 

 But for the large values of r, <r x is equal to what a' becomes when 



ir is put for 0. Hence ctj= -ysin q(a't + c ). 



As the forms of the functions / and F depend entirely on that 

 of the function which expresses the law of the velocity and con- 

 densation of the incident undulations, it will be assumed that 



f=m l sin^(r— a't + c^), ~F = m 2 sin q(r + a' t + c 2 ). 



