acted upon by the Undulations of an Elastic Fluid. 355 

 By the condition that U = where r~c } we have 

 2(/i-g.) »(/-F) ifl F ^ 0i 



After substituting in this equation the above values of/ and F 

 and putting c for r, it will be found that the equation is satisfied 

 for all values of t if m 2 = —m x and c 2 =c v and if the arbitrary 

 constant c x is determined by the equation 



tan ? ( C + Cl )=-l + | C . 



Also if we take another set of values of/ and F, distinguishing 

 them from the preceding by dashes attached to the constants, 

 the same equation will be satisfied if m' 2 =m' 1 and c' 2 =c'j, and 

 if c\ be determined by the equation 



cotg'(c + c' 1 ) = ~- 



2V u qc 2 



As these two methods of satisfying the condition U = are 

 equally entitled to consideration, both must be employed in de- 

 ducing the value of a. Here it is to be remarked that, on account 

 of the linear form of the differential equation from which <f>^ is 

 obtained, we might have 2/ and 2F in place of/ and F. This 

 being the case, it is allowable to substitute in the expression for 

 cr — a 1 the respective sums of the two values of/ and F. When 

 this is done, and the relations between the constants are taken 

 into account, the result is 



(2m, , v 2m, q , . \ . . n 



— g- 1 cos^r + Cj) H — sm q(r + Cj) 1 sm qa't cos a 



— ('-—sin q{r + c\) —cos q{r-\-c'^) J cos qa't cos 0. 



At the same time the foregoing equations for finding Cj and c', 

 give very approximately 



cos qc l = — -^— > singc£=l; cosqc' i = l ) sin qc' 1 =^— 



By having regard to these values of c x and c\, expanding the 

 sines and cosines, and omitting insignificant terms, the above 

 equation becomes 



a— a l — — I \-^ H — q— ) ( m \ s ^ n Q.dt + m\ cos qa't) cos 6. 



When r is very large compared to c, the first term in the first 

 brackets may be omitted, and the consequent value of <r — cr, 

 must then satisfy the condition of being identical with the term 



