302 I>R. R- CHAPMAN ON THE DIURNAL VARIATIONS OF THE 



is clear that by altering the constant term in p, so that p never falls to zero, the 

 ;I|K>V<- calculations become formally and really legitimate ; when we wish to return to 

 the actual case we must appeal to the " law of continuity," and the fact that our 

 mathematics is applied to an ordinary physical problem, to allow us to pass to the 

 limiting value of d a in the final result. The latter is expressed as a power series 

 in \jd , and if d is sufficiently diminished, this series might become non-convergent. 

 But the actual results do not indicate any such behaviour, and are, as we have seen, 

 identical with those obtained by SCHUSTER'S method (in which the conductivity only 

 was considered), so far as the scope of the two calculations is the same. 



23. So far the calculations have been kept quite general, in that no relation 

 between the causes of the variable conductivity and of the atmospheric oscillation 

 has been assumed. Thus they may both be caused by the sun, in which case the 

 mathematics is that applicable to the theory of the solar diurnal variations of the 

 earth's magnetism. Without much modification, however, they may equally well be 

 adapted to the case of the lunar diurnal variations. We shall consider it sufficient, 

 for our purpose, to regard the solar and lunar periods as equal at any one time, 

 allowing for the slow cumulative effect of their inequality by introducing a variable 

 phase angle v into the expression for cos w, the quantity on which p and K depend. 

 Thus 



cos to = sin <5 cos + cos<?sin0 cos (\ + t' + v), 



where tf is now the local lunar time of the standard meridian (measured from upper 

 culmination), and v measures the lunar phase, increasing from to 2-rr from one new 

 moon to the next. The velocity potential will be Q 2 2 sin (2\ + t'a). The calculations 

 will be formally the same if we now change the meaning of X' to X + t'+v, so that the 

 velocity potential becomes 



Q/sin(2X'-a-2i>). 



Thus by equation (6) the current function obtained is 



2 J>/Q/sin{rA'-a-2,-} + 2 q n 'QS tun(a\'+a + 2v) 



= <r=l 



2 g/Q/sin {<r(\ + ') + + (<r+ 2)*}. 



<r=l 



The terms on the left of the last line change in phase through an angle 2(o 2)v 

 each month, viz., -2* for the diurnal term, zero for the semi-diurnal term, 

 + 2ir for the third component, and +4ir for the fourth component, as the observa- 

 tions indicated. The terms on the left change phase by 2(^+2)^ each month, 

 a change so rapid that it would be difficult to detect in the observations, 

 affected as these are by accidental error. The coefficients q n ', moreover, are very 

 small, so that altogether these terms are negligible. 



One interesting result of the analysis may be noticed here, viz., that the main 



