400 Sir William Thomson on the General Integration 



that this transcendental equation cannot have imaginary roots 

 and has necessarily an infinite number of real roots more and 

 more nearly equidifferent when taken in order of magnitude 

 from the smallest positive to larger and larger positive, or from 

 the smallest negative to larger and larger negative. In the 

 case of 5 = the positive and negative roots are equal, unequal 

 in all other cases (s = l, 5 = 2, &c). 



15. For the convergency of the series in (50) and (51) it is 

 necessary and sufficient (§ 9) that there be no root, real or 

 imaginary, of y = whose absolute magnitude is less than that 

 of the absolutely greater of the two quantities p! and pi 1 . But 

 it is only when, with algebraic signs taken into account, there 

 is a real root actually between pi and p," (that is to say, when 7 

 becomes zero for some value of p, on the direct range from pi to 

 pi') that any of the six functions a (/a), fi(p), X{pl) } ci{p>), fi(p>)j 

 \(pl) used in the processes (53), (54), (57), (58), and in the 

 final solution (51), (50), (55), is discontinuous. Why some or 

 all of these functions should be discontinuous in this case is 

 obvious : the sea's depth being zero along any parallel of 

 latitude limits the physical problem to the side on which the 

 depth is positive, or (case of equal roots of 7=0) separates the 

 problem into two independent ones, to find the motions of the 

 water on the two sides of a reef just " awash ." An imaginary 

 root of 7=0 having its absolute magnitude II between pi and 

 ///', ,ov a real root of contrary sign to the absolutely greater of 

 pi and ///', and of absolute magnitude R between them, renders 

 the series for <*(p>), fi(p)> &c. in ascending powers of p, diver- 

 gent for the portion of our range of latitude which lies beyond 

 + sin -1 R. Still the solution of the problem is fully given by 

 (55) in terms of six functions a (p,), 0(p), &c, each continuous 

 throughout the range, but calculable, by the series in ascending 

 powers of p. set forth in our preceding formulae, only for the 

 part of the range of latitude which lies between —sin -1 R and 

 + sin -1 R. The mode of dealing with the case of imaginary roots 

 so as to obtain convenient formulae for the numerical calculation 

 of ct(p) &c. is an interesting and important subject to which I 

 hope to return. Being (§ 9) at present limited to the case of 

 real roots, it is enough to remark that in this for each of the 

 six functions a (/jl), ft{p), &c. a series continuously convergent 

 throughout the range from p! to pi' may be found thus : — Let 

 p and p 1 be consecutive real roots of 7 = 0, and let p, pi, pl' } p l 

 be in order of algebraic magnitude. Let a be any quantity such 

 that algebraically 



and V (59) 



