r.AKTII'S MACNKT1SM I'KODITKD BY THE MOON AND SUN. 295 



to leave the type unchanged, while extending the range of the degrees by 4q. Also 

 by equation (A), the coefficient a, which, it will be remembered, is a power series in 

 cos 0, leaves the type unchanged while it increases the range of the degrees of the 

 resulting tesseral harmonics. In every case, therefore (p positive or negative), It/ 

 can be expressed as the sum of a number of terms such as Q,**'. Therefore if we 

 write ^ for the sum of all the expressions (2) resulting from each term Q m T sin (T\' a) 

 in the velocity potential \}s, the fundamental equation (l) for R takes the form 



(4) ^ = 2 { 2 k n '- p Q n '* f sin (<r+p . \' a')}, 



where k n '' p is a coefficient whose value can be determined in terms of p n * and the 

 coefficients d p in the TAYLOR'S series for K. By equating the coefficients of harmonics 

 of the same degree and type, on the two sides of the equation, we obtain equations 

 to determine the coefficient p m ' in terms of the d p and the known constants of 

 the velocity potential. In practice this must be done by a process of successive 

 approximation. Knowing, from the form of the above equation, which is linear in 

 p" and d p , that every coefficient p," can be expressed as a TAYLOR'S series in 



-^ , -^ , -j 2 , and so on, we can determine this series by successively assuming that 



all save one particular variable -f are zero, and considering this variable alone, it 



may easily l>e seen that the phase angle of every term in R arising from a particular 

 term in Sk is the same as that of the latter. 



18. SCHUSTER has worked out the values of p n ' for the special form of conductivity 

 already mentioned, and for the two terms Q, 1 sin (\' a) and Q/sin(2\' a) in the 

 velocity potential, to the fourth order of approximation, and he finds that the 

 numerical coefficients of the terms are such that only the first order term (depending, 

 in our notation, on d t /d a ) are large enough to be detectable by olwervation. The 

 present calculation will not be carried so far, therefore, and will not include terms of 

 higher order than d.Jd or (di/d ) 3 . Also, since in the expression for ^ the term 

 depending on the inclination of the magnetic to the geographical axis is multiplied 

 by the small factor tan <f>, the part of It depending on this term will only be calculated 

 as far as the first order d t fd . Further, since the actual atmospheric oscillations seem 

 to be mainly performed in the simplest mode possible, so that m = r for the principal 

 terms, the second order terms will be neglected for the smaller harmonics in the 

 velocity potential \js, for which in ^ T. 



We therefore consider the terms in R which depend upon a term A M ' T Q m T in "b, 

 where m' and r are quite general, except that in the terms depending on d-Jd^ we 

 shall suppose m' = r'+l (since Q,/ = when T > m', the term in (2) depending on 

 Q*-i T vanishes when m = r). 



It will first be necessary to write out the developed expressions for R/(o), 

 R/(l)> IV (2) as far as the terms in d a . No other values of p in R,*(p) give 



