MAONI-TISM PRODUCED BY THE MOON AND SUN. 289 



if <f is the dtvlinutinn of the sun, ev'ulmtly we have 



cos = sin & cos 6+ cos tJsin 



\vliriv 



x = sin S cos 0, 1y = cos ^ sin 0, /u = cos (\ + 1). 



The conductivity and resistivity at the point (0, \) will be denoted by p and ' 

 respectively, p' being, of course, equal to unity. For the present we shall suppose 

 that p and / are finite and continuous functions of to, so that they can be expressed 

 as FOURIER'S series in cos nu> over the range 0, v ; p will certainly satisfy this 

 condition, and the case of p = 0, K = will be considered later. Further, it will be 

 assumed possible to express K' as a TAYLOR'S series in cos w, and it is in this form that 

 we shall suppose the resistivity to be given, as one of the data of the problem. 

 Theoretically this is a limitation of the problem, as there are some functions which 

 cannot be expressed in the form stated ; for instance, if the conductivity were 

 proportional to cosco in that hemisphere on which the sun is shining, and zero or 

 constant over the other hemisphere, K' could not be so expressed. But in reality 

 nothing of value is lost, as any continuous function can be approximately expressed 

 in the form of a TAYLOR'S series to any desired degree of accuracy. 



[Some further explanation of this use of series may be desirable. The series used 

 in the analysis are all written as infinite ones, for the sake of formal simplicity and 

 theoretical completeness. In the detailed execution of the work, however, only a 

 finite number of these terms can be utilized, as workable general expressions for the 

 coefficients in the current function II cannot be obtained. The actual procedure, 

 therefore, must be to take a finite number of terms of the FOURIER'S series for />, 

 transform this into a polynomial in cos u> (this also, of course, will have only a finite 

 number of terms), and work out the coefficients of R in terms of the coefficients of 

 this polynomial to as great a degree of accuracy as is practicable and desirable. 

 This is the course of the work in 18-20, where the terms (a + b cos u> + c cos 2o>) of 

 the FOURIER'S series for p are taken, and the expression for R is worked out as far as 

 concerns the terms in a, b, />", and c. The resistivity ifp is introduced into the 

 calculations for purely mathematical reasons, on account of certain analytical 

 advantages which it seems to offer. The results obtained in this way, in terms of 

 the coefficients of p, might be got otherwise by an extension of the method used by 

 SCHUSTER. This identity of results is clear from the fact that if the FOURIER 

 coefficients of p are small enough the TAYLOR'S series for l/p is absolutely convergent, 

 and the legitimacy of the use of l/p is in this case immediately evident ; the formal 

 results, however, do not depend on any property of convergence, so that the results 

 obtained by using l/p remain equally valid with those obtained in any other way, 

 even though the series for l/p should become non-convergent. This is one of many 

 instances in which it is possible and advantageous to use expressions which may 



VOL. ccxin. A. _' i 1 



