of Laplace's Differential Equation of the Tides. 397 



where a, b, c, d, e } f denote constants. The complete solution of 

 this approximate equation may be found by assuming 



0= log *(H + H l * + H a ** + &c.)l m 



and determining H l5 H 2 , &c. in terms of H , arbitrary, by equa- 

 ting coefficients of log,?, zlogz, z 2 iogz, &c. to zero, and lastly 

 determining K v K 2 , K 3 in terms of K and H , each arbitrary, 

 and Hj, H 2 , H 3 , &c. previously found. This shows the kind of 

 discontinuity which any complete solution of the exact equation 

 (14) necessarily presents- when the value of fi passes through a 

 real root of y = 0, and how this discontinuity is averted by an 

 assignment of the two constants of integration in the rigorous 

 solution proper to make H =0 in the approximate solution (45). 

 11. Return now to the question (§ 9) of assigning the two 

 constants of integration so as to fulfil any proper physical con- 

 ditions of our problem. First; to work out the general solution in 

 ascending powers of fju, use (40), and calculate K 1; K 2 , &c. suc- 

 cessively with C and K arbitrary. Thus we find 



K i ==C« i + K A+^° ) + ^ 1) + ^ 2) + &c., . (47) 



where ct h ft, \$°\ \P, xf>, &c. are numbers calculated by the 

 process, supposing/, s, m, y 0i y v 7 2 , &c. to have had any parti- 

 cular numerical values assigned to them, and <E> , <£„ <E> 2 > . . . to 

 denote given heights. Or if before we begin the arithmetical 

 process particular values are assigned to <E> , <& u *I> 2 , &c. so that 

 we may put 



<£> = ?z L, ^ l — n l li ) <f> 2 =n 2 L, &c, . . . (48) 



L denoting a given line, and n Q) n v n q , &c. given numerical 

 quantities, the result of the process of calculation of K f from (40) 



K^a.B+ftKo + ^L, (49) 



where u i} ft, \ are calculated numbers. Then we have, by (15) 

 and (16), 



j i ^ ? *=«W.C+i8( f *).K +XW.L J 



where a( / c6)=a + « 1 /Lt + a 2/ u, 2 + &c, 



\{fi) =X + \fi + \fi z + &c., 

 and 



